WEBVTT 1 00:00:00.190 --> 00:00:00.670 Christophe Raymond Garban: Good. 2 00:00:02.250 --> 00:00:06.270 Christophe Raymond Garban: Well, everyone's environment is… 3 00:00:06.520 --> 00:00:07.930 Courant Events Right: And beaches. 4 00:00:09.400 --> 00:00:12.649 Courant Events Right: It's been systems with continuous symmetries. 5 00:00:12.770 --> 00:00:14.000 Courant Events Right: Okay, Jill, I agree. 6 00:00:14.670 --> 00:00:21.890 Courant Events Right: Okay, so, thank you very much, Rollo and Lina, for the invitation. It's a pleasure to speak in front of… 7 00:00:22.070 --> 00:00:24.780 Courant Events Right: As many mathematicians and physicists. 8 00:00:25.290 --> 00:00:33.250 Courant Events Right: So, as asked by Ronan and Nina, I will split the talk in two parts. The first part will be a review 9 00:00:33.410 --> 00:00:37.179 Courant Events Right: Of spin systems which have a continuous symmetry. 10 00:00:37.390 --> 00:00:42.710 Courant Events Right: So this will be the first part, and the second part, I will tell you, it will be a bit of rooftop. 11 00:00:43.270 --> 00:00:43.780 Christophe Raymond Garban: MBL. 12 00:00:43.780 --> 00:00:52.109 Courant Events Right: maybe, for the future. White, lemon, orange, and flat peach, follow different trajectories under the RG flow. 13 00:00:53.410 --> 00:00:56.930 Christophe Raymond Garban: So, first part, first, so… I won't. 14 00:00:56.930 --> 00:01:03.359 Courant Events Right: to tell you about, spin systems which live on a lattice, ZD. 15 00:01:03.430 --> 00:01:05.670 Christophe Raymond Garban: But in this talk, almost. 16 00:01:05.670 --> 00:01:11.499 Courant Events Right: all the time, dimension will be D is equal to 2, so it will be the Z2 grid. 17 00:01:11.850 --> 00:01:18.660 Courant Events Right: And each point in this Z2 grid will carry a spin. Only the target space will change. 18 00:01:18.910 --> 00:01:21.429 Courant Events Right: In the first picture here, the target space. 19 00:01:21.430 --> 00:01:22.069 Christophe Raymond Garban: This is blessed. 20 00:01:22.070 --> 00:01:25.570 Courant Events Right: minus 1. The underlying symmetry is discrete. 21 00:01:25.740 --> 00:01:30.309 Courant Events Right: And it's, I think, well known to everyone in the room. It's called the easing model. 22 00:01:30.710 --> 00:01:35.740 Courant Events Right: The second example here, which will be related to the beach later on. 23 00:01:35.930 --> 00:01:43.710 Courant Events Right: is the XY model, so here, every point on Z2 carries a spin in the unit circle S1. 24 00:01:44.140 --> 00:01:46.000 Courant Events Right: And this last example, which. 25 00:01:46.150 --> 00:01:46.729 Christophe Raymond Garban: Plenty of. 26 00:01:46.730 --> 00:01:48.229 Courant Events Right: One will be the orange. 27 00:01:48.360 --> 00:01:51.619 Courant Events Right: It's the case where every spin carries an S2. 28 00:01:51.620 --> 00:01:54.360 Christophe Raymond Garban: spin, and it's called the classical Eisenh. 29 00:01:54.360 --> 00:01:55.260 Courant Events Right: their 30 00:01:56.020 --> 00:02:02.379 Courant Events Right: The good news with these three models is that they share the same lattice, and they also share the same Gibbs measure. 31 00:02:02.560 --> 00:02:08.430 Courant Events Right: Each configuration of spin sigma, whatever the target space is here, will be the. 32 00:02:08.430 --> 00:02:11.780 Christophe Raymond Garban: Guarantee will be proportional to exponential beta. 33 00:02:11.920 --> 00:02:12.430 Christophe Raymond Garban: Tony? 34 00:02:12.430 --> 00:02:22.569 Courant Events Right: a sum of our edges, i next to J, so this is an edges of Z2, this is another edge of Z2, and each of these edges will carry the. 35 00:02:22.570 --> 00:02:24.229 Christophe Raymond Garban: product sigma i times. 36 00:02:24.230 --> 00:02:24.920 Courant Events Right: Sigma J. 37 00:02:25.400 --> 00:02:25.970 Christophe Raymond Garban: So that's… 38 00:02:25.970 --> 00:02:26.969 Courant Events Right: So when beta is large. 39 00:02:26.970 --> 00:02:29.390 Christophe Raymond Garban: The system will be… will look very ordered. 40 00:02:29.390 --> 00:02:31.389 Courant Events Right: At least in a large enough window. 41 00:02:31.910 --> 00:02:37.170 Christophe Raymond Garban: And we'll see that if we maybe want to arbitrarily large windows, in this case here, the order. 42 00:02:37.170 --> 00:02:38.689 Courant Events Right: Or maybe completely lost. 43 00:02:39.390 --> 00:02:45.020 Courant Events Right: An equivalent way to consider these spin systems is to… the Gibbs measure is also exactly. 44 00:02:45.020 --> 00:02:45.890 Christophe Raymond Garban: Exactly, experimentally. 45 00:02:45.890 --> 00:02:47.420 Courant Events Right: And in short, minus beta. 46 00:02:47.420 --> 00:02:48.420 Christophe Raymond Garban: or the two. 47 00:02:48.740 --> 00:02:49.230 Courant Events Right: This time. 48 00:02:49.230 --> 00:02:50.170 Christophe Raymond Garban: sum over N. 49 00:02:50.170 --> 00:02:55.950 Courant Events Right: is not the Scala product, but an energy, it would be the D.ichlet energy along the edge. 50 00:02:56.170 --> 00:02:58.690 Courant Events Right: I will sum over all the edges of the 51 00:02:59.460 --> 00:03:01.290 Courant Events Right: The gradient of the spin squared. 52 00:03:03.040 --> 00:03:03.570 Christophe Raymond Garban: So… 53 00:03:03.570 --> 00:03:04.979 Courant Events Right: Depending on your viewpoint. 54 00:03:04.980 --> 00:03:05.910 Christophe Raymond Garban: Either you won't. 55 00:03:05.910 --> 00:03:08.779 Courant Events Right: to maximize the sum of scalar product. 56 00:03:08.980 --> 00:03:09.450 Christophe Raymond Garban: Or, maybe. 57 00:03:09.450 --> 00:03:18.180 Courant Events Right: Maybe, if you're a physicist, you want to minimize the energy, so you want to minimize the dirichlet energy you pay over each edge of your graph. 58 00:03:19.650 --> 00:03:24.450 Courant Events Right: Okay, so… That's the three spin systems on the. 59 00:03:24.450 --> 00:03:25.680 Christophe Raymond Garban: I mentioned, too, that I will. 60 00:03:25.680 --> 00:03:27.669 Courant Events Right: Beta will always be the usual. 61 00:03:27.670 --> 00:03:29.580 Christophe Raymond Garban: inverse temperature. 62 00:03:30.570 --> 00:03:33.239 Courant Events Right: So, what I will do… 63 00:03:33.240 --> 00:03:35.819 Christophe Raymond Garban: in this first part of the talk is, for each of. 64 00:03:35.820 --> 00:03:37.909 Courant Events Right: these models, I would want to give you 65 00:03:38.200 --> 00:03:41.679 Courant Events Right: The way physicists think about the spin system, or at least. 66 00:03:41.680 --> 00:03:44.480 Christophe Raymond Garban: way, I think I understood what physicists. 67 00:03:44.480 --> 00:03:45.940 Courant Events Right: Trying to do with this. 68 00:03:45.940 --> 00:03:46.429 Christophe Raymond Garban: It's so much. 69 00:03:46.430 --> 00:03:48.660 Courant Events Right: try to translate the way I understood. 70 00:03:48.660 --> 00:03:50.220 Christophe Raymond Garban: Completely misunderstood. 71 00:03:50.750 --> 00:03:58.960 Courant Events Right: I will also try to give you the mathematical way, and I will proceed as follows. For the easing model, I will only give you the physics way. 72 00:03:58.960 --> 00:03:59.730 Christophe Raymond Garban: But. 73 00:03:59.730 --> 00:04:00.939 Courant Events Right: The masro is extremely close. 74 00:04:00.940 --> 00:04:02.380 Christophe Raymond Garban: Northwestern, all of what. 75 00:04:02.380 --> 00:04:03.060 Courant Events Right: I'm going to do. 76 00:04:03.060 --> 00:04:06.690 Christophe Raymond Garban: The boat would be 2 minutes long and almost 100 years old. 77 00:04:07.400 --> 00:04:13.100 Courant Events Right: For the XY model, I insist I will give the two ways, and there, they are quite different from each other. 78 00:04:13.530 --> 00:04:14.050 Christophe Raymond Garban: I will. 79 00:04:14.050 --> 00:04:14.640 Courant Events Right: give you the. 80 00:04:14.640 --> 00:04:16.250 Christophe Raymond Garban: The way business. 81 00:04:16.250 --> 00:04:22.179 Courant Events Right: have thought about it, and then I will give you the way Frollish and Spencer have thought about it. 82 00:04:22.630 --> 00:04:31.760 Courant Events Right: For the last model here, I wish I would give you a physics and a math way, but I will only be able to give you a physics way. The math way is lacking so far. 83 00:04:33.210 --> 00:04:34.200 Christophe Raymond Garban: So, first. 84 00:04:34.200 --> 00:04:36.680 Courant Events Right: The easing model, so that's the case where. 85 00:04:36.680 --> 00:04:40.279 Christophe Raymond Garban: There is a discrete symmetry, so that's not supposed. 86 00:04:40.280 --> 00:04:43.720 Courant Events Right: to be the overview Roland and Nina asked about, so there's no underlying. 87 00:04:43.720 --> 00:04:44.510 Christophe Raymond Garban: buying continuous. 88 00:04:44.510 --> 00:04:45.460 Courant Events Right: Symmetry in the. 89 00:04:45.460 --> 00:04:47.899 Christophe Raymond Garban: target space. Still, it's, of course. 90 00:04:47.900 --> 00:04:52.420 Courant Events Right: the next model, and that's a simulation stolen from the web. 91 00:04:52.800 --> 00:04:53.340 Christophe Raymond Garban: Where… 92 00:04:53.340 --> 00:05:00.349 Courant Events Right: I have a high temperature easing model here. I have the exactly critical easing model here with its nice fractal structures. 93 00:05:00.970 --> 00:05:01.549 Christophe Raymond Garban: And when. 94 00:05:01.550 --> 00:05:03.940 Courant Events Right: When beta is large, the system picks. 95 00:05:03.980 --> 00:05:07.079 Christophe Raymond Garban: A color, plus or minus, black or white. 96 00:05:07.320 --> 00:05:08.680 Christophe Raymond Garban: Things to eat, and there are. 97 00:05:08.680 --> 00:05:11.269 Courant Events Right: Only slow, villages. 98 00:05:11.270 --> 00:05:12.219 Christophe Raymond Garban: Resisting to this. 99 00:05:12.220 --> 00:05:14.269 Courant Events Right: ground states, long-range order. 100 00:05:15.260 --> 00:05:16.490 Christophe Raymond Garban: So, just for. 101 00:05:16.490 --> 00:05:27.679 Courant Events Right: a couple of minutes. It's a little bit risky for me, but I will try to give you maybe the way physicists think about it. If you think that it's completely wrong, please tell me after the talk. 102 00:05:27.680 --> 00:05:30.240 Christophe Raymond Garban: So that they keep some… some… 103 00:05:30.240 --> 00:05:31.230 Courant Events Right: visibility. 104 00:05:31.490 --> 00:05:32.090 Christophe Raymond Garban: So… 105 00:05:32.090 --> 00:05:33.020 Courant Events Right: So, 106 00:05:33.020 --> 00:05:33.819 Christophe Raymond Garban: I will hide it. 107 00:05:33.820 --> 00:05:37.519 Courant Events Right: Mayor's argument with some words like excitations and things. 108 00:05:37.520 --> 00:05:38.260 Christophe Raymond Garban: like this. 109 00:05:38.260 --> 00:05:46.809 Courant Events Right: This is a… imagine a huge piece of 2D lattice with a way to break the symmetry outside, which would be a plus boundary condition outside. 110 00:05:46.810 --> 00:05:48.179 Christophe Raymond Garban: the blue condition. 111 00:05:48.410 --> 00:05:54.680 Courant Events Right: And now, we want to understand how the easing model is reacting to temperature, to the fact beta. 112 00:05:54.840 --> 00:05:56.830 Christophe Raymond Garban: is not infinite temperature T. 113 00:05:56.830 --> 00:05:57.799 Courant Events Right: P is non-zero. 114 00:05:57.980 --> 00:06:00.089 Christophe Raymond Garban: And there are thermal fluctuations. 115 00:06:00.710 --> 00:06:06.439 Courant Events Right: how to preempt this? Maybe there is this ground state, which is the blue ground state, reacting to the bonding. 116 00:06:06.870 --> 00:06:07.620 Christophe Raymond Garban: And now… 117 00:06:07.620 --> 00:06:12.359 Courant Events Right: There are some possible excitations which will break this, this bronze 118 00:06:13.320 --> 00:06:16.800 Courant Events Right: This excitation could be represented by a loop, a closed-loop. 119 00:06:16.800 --> 00:06:20.130 Christophe Raymond Garban: And we can see that one good loop has. 120 00:06:20.130 --> 00:06:24.540 Courant Events Right: a certain energy associated to it, and it's essentially beta. 121 00:06:24.540 --> 00:06:25.610 Christophe Raymond Garban: Sometimes the instant. 122 00:06:25.610 --> 00:06:27.809 Courant Events Right: So, essentially, the length of the loop. 123 00:06:28.040 --> 00:06:34.970 Courant Events Right: And so the energy of one such excitation in the Gibbs state is exponential minus beta times this length. 124 00:06:35.440 --> 00:06:40.100 Courant Events Right: The entropy is the number of such excitations of length n 125 00:06:40.220 --> 00:06:44.559 Courant Events Right: And there is a classical fight between energy and entropy. 126 00:06:44.990 --> 00:06:45.539 Christophe Raymond Garban: I insist. 127 00:06:45.540 --> 00:06:48.819 Courant Events Right: on these arguments, because I will use the same argument 128 00:06:49.140 --> 00:06:51.610 Courant Events Right: for the phase transition for the X1. 129 00:06:51.610 --> 00:06:52.310 Christophe Raymond Garban: Family. 130 00:06:52.850 --> 00:06:54.280 Christophe Raymond Garban: So, if beta is large. 131 00:06:54.280 --> 00:06:55.129 Courant Events Right: You pay a lot. 132 00:06:55.130 --> 00:06:55.640 Christophe Raymond Garban: of energy. 133 00:06:55.640 --> 00:06:56.800 Courant Events Right: And the effect of this. 134 00:06:56.800 --> 00:06:57.340 Christophe Raymond Garban: This is dead. 135 00:06:57.340 --> 00:06:58.580 Courant Events Right: You need for me in the size. 136 00:06:58.580 --> 00:06:59.680 Christophe Raymond Garban: of the system. 137 00:07:00.030 --> 00:07:15.590 Courant Events Right: Of course, there will be some excitations here and there, but the probability to see a massive excitation of the scale of your domain can easily be seen to be very unlikely. So you will only see rather small excitations, and in the bulk of the system. 138 00:07:15.590 --> 00:07:16.270 Christophe Raymond Garban: him. 139 00:07:16.270 --> 00:07:19.069 Courant Events Right: The ground state will still be there, and you will still. 140 00:07:19.070 --> 00:07:20.180 Christophe Raymond Garban: bring this imagery. 141 00:07:20.790 --> 00:07:22.849 Christophe Raymond Garban: As beta gets lower. 142 00:07:22.850 --> 00:07:23.220 Courant Events Right: the… 143 00:07:23.220 --> 00:07:24.409 Christophe Raymond Garban: Zvich is large. 144 00:07:24.410 --> 00:07:25.890 Courant Events Right: and temperature gets lower. 145 00:07:25.890 --> 00:07:27.250 Christophe Raymond Garban: lower, 146 00:07:27.710 --> 00:07:28.989 Courant Events Right: No, the opposite way. 147 00:07:28.990 --> 00:07:30.490 Christophe Raymond Garban: always confuse beta and T. 148 00:07:30.490 --> 00:07:34.109 Courant Events Right: As temperature gets higher, as beta gets lower. 149 00:07:34.520 --> 00:07:35.390 Christophe Raymond Garban: Excellent. 150 00:07:35.390 --> 00:07:37.300 Courant Events Right: are less costly in terms of the. 151 00:07:37.300 --> 00:07:42.030 Christophe Raymond Garban: energy. Entropy is still there, and so after a while, you start having. 152 00:07:42.030 --> 00:07:42.470 Courant Events Right: more and more. 153 00:07:42.470 --> 00:07:44.820 Christophe Raymond Garban: More excitations piling up on each other. 154 00:07:45.390 --> 00:07:49.179 Christophe Raymond Garban: In such an efficient way that in the bulk of the system, you completely forgot. 155 00:07:49.180 --> 00:07:49.810 Courant Events Right: good about. 156 00:07:49.810 --> 00:07:50.660 Christophe Raymond Garban: the symmetry. 157 00:07:50.660 --> 00:07:52.150 Courant Events Right: breaking, you applied at the. 158 00:07:52.150 --> 00:07:52.820 Christophe Raymond Garban: on there. 159 00:07:53.200 --> 00:07:54.609 Christophe Raymond Garban: So this is, I think. 160 00:07:54.610 --> 00:08:00.420 Courant Events Right: think the way physicists may explain easing model, though I could be completely wrong here. 161 00:08:00.850 --> 00:08:06.560 Courant Events Right: So, I want to move now to continuous symmetry, and this was the second example, the. 162 00:08:06.560 --> 00:08:09.819 Christophe Raymond Garban: example in my talk, where each point in Z2. 163 00:08:09.820 --> 00:08:12.800 Courant Events Right: So this is a 500, 500 pictures stolen from. 164 00:08:12.800 --> 00:08:13.619 Christophe Raymond Garban: Give me the… 165 00:08:14.130 --> 00:08:19.469 Courant Events Right: each point carries a spin in the unit circle S1, and this, pixel. 166 00:08:19.470 --> 00:08:22.490 Christophe Raymond Garban: That is colored, according to the angle here. 167 00:08:23.030 --> 00:08:27.529 Courant Events Right: If the temperature t is very large, if beta is very small, this is a very busy. 168 00:08:27.530 --> 00:08:28.669 Christophe Raymond Garban: problems being true. 169 00:08:28.850 --> 00:08:30.070 Christophe Raymond Garban: Well, if the temperature. 170 00:08:30.070 --> 00:08:36.619 Courant Events Right: is low. This looks like we have more structure coming into the picture, but still, there is no dominant. 171 00:08:36.620 --> 00:08:37.479 Christophe Raymond Garban: There's no domin. 172 00:08:37.480 --> 00:08:37.799 Courant Events Right: England. 173 00:08:37.809 --> 00:08:38.409 Christophe Raymond Garban: angle with. 174 00:08:38.409 --> 00:08:39.379 Courant Events Right: small fluctuation. 175 00:08:39.380 --> 00:08:40.070 Christophe Raymond Garban: around the. 176 00:08:41.080 --> 00:08:51.239 Courant Events Right: And this is for a well-known reason. This goes back to a theorem by Mermin and Wagner in the 60s, and what they've shown is that in demonstrating. 177 00:08:51.240 --> 00:08:51.809 Christophe Raymond Garban: is equal to. 178 00:08:51.810 --> 00:08:58.510 Courant Events Right: If you take any spin system which has a continuous symmetry, it could be XY, it could be. 179 00:08:58.510 --> 00:08:59.450 Christophe Raymond Garban: Hi, man. 180 00:08:59.710 --> 00:09:04.200 Courant Events Right: Or it could be a Lee group-valued spin system with a continuous symmetry there. 181 00:09:04.220 --> 00:09:04.930 Christophe Raymond Garban: Like. 182 00:09:04.930 --> 00:09:06.050 Courant Events Right: SO3, for example. 183 00:09:06.710 --> 00:09:07.970 Christophe Raymond Garban: Then. 184 00:09:08.840 --> 00:09:13.340 Courant Events Right: They've shown that it's impossible to break the symmetry. 185 00:09:13.530 --> 00:09:18.060 Courant Events Right: And the main argument they relied on is called an harmonic map. 186 00:09:18.430 --> 00:09:19.390 Christophe Raymond Garban: It's appropriate. 187 00:09:19.390 --> 00:09:19.730 Courant Events Right: he was. 188 00:09:19.730 --> 00:09:20.579 Christophe Raymond Garban: in, 189 00:09:20.580 --> 00:09:22.920 Courant Events Right: Harmonic analysis, which is such that if you. 190 00:09:22.920 --> 00:09:25.309 Christophe Raymond Garban: I'm trying to break the symmetry outside. 191 00:09:25.540 --> 00:09:28.649 Courant Events Right: For example, here with the east pointing direction. 192 00:09:28.990 --> 00:09:29.520 Christophe Raymond Garban: And you. 193 00:09:29.520 --> 00:09:37.079 Courant Events Right: Ask yourself, how is it energetically costly to point in some other direction in the bulk of the system? 194 00:09:38.340 --> 00:09:39.050 Courant Events Right: I found out. 195 00:09:39.050 --> 00:09:41.570 Christophe Raymond Garban: That there is a deterministic. 196 00:09:41.570 --> 00:09:44.819 Courant Events Right: of angles in this end by inbox, which is such that. 197 00:09:45.130 --> 00:09:45.640 Christophe Raymond Garban: You can… 198 00:09:45.640 --> 00:09:54.499 Courant Events Right: slowly interpolate from one to the other in such a way that you only pay a dirichlet energy, which vanishes to zero as the. 199 00:09:54.500 --> 00:09:55.659 Christophe Raymond Garban: Because of this domain. 200 00:09:55.660 --> 00:09:56.520 Courant Events Right: It grows to infinity. 201 00:09:57.630 --> 00:09:58.510 Christophe Raymond Garban: If you want to. 202 00:09:58.510 --> 00:10:00.449 Courant Events Right: to do the exercise and get bored at some point. 203 00:10:00.870 --> 00:10:03.440 Courant Events Right: If you linearly interpolate at linear speed. 204 00:10:03.440 --> 00:10:04.190 Christophe Raymond Garban: on the Pandailles. 205 00:10:04.190 --> 00:10:08.760 Courant Events Right: to the middle, this will not go to zero. So you need to do something a little bit. 206 00:10:08.760 --> 00:10:09.640 Christophe Raymond Garban: Smarter. 207 00:10:09.640 --> 00:10:10.160 Courant Events Right: Thank you. 208 00:10:10.980 --> 00:10:16.909 Courant Events Right: And since it costs nearly zero energy to forget about the Pandari direction. 209 00:10:17.250 --> 00:10:22.130 Christophe Raymond Garban: They, could show that there is no phase… no symmetry breaking for. 210 00:10:22.130 --> 00:10:23.900 Courant Events Right: these systems in Dimension 2. 211 00:10:24.360 --> 00:10:26.140 Courant Events Right: And I think I was not today. 212 00:10:26.140 --> 00:10:28.119 Christophe Raymond Garban: In the 60s, but. 213 00:10:28.410 --> 00:10:31.570 Courant Events Right: I've heard some videos At the time. 214 00:10:31.570 --> 00:10:32.290 Christophe Raymond Garban: There's also. 215 00:10:32.290 --> 00:10:37.679 Courant Events Right: okay, probably XY model, classical Eisenberg, all of those in dimension 2, they're pretty. 216 00:10:37.680 --> 00:10:38.590 Christophe Raymond Garban: boring, because there's. 217 00:10:38.590 --> 00:10:39.320 Courant Events Right: know. 218 00:10:39.510 --> 00:10:40.010 Christophe Raymond Garban: bring. 219 00:10:40.010 --> 00:10:46.100 Courant Events Right: of symmetry And it was maybe considered a little bit like a one-dimensional easing model. 220 00:10:46.730 --> 00:10:55.079 Courant Events Right: That was until the early 70s, when Berzynski, Kostoletz, and Taurus Phone that's, basically. 221 00:10:55.080 --> 00:10:58.239 Christophe Raymond Garban: sites, this serum by Mamine and Wagner. Still. 222 00:10:58.240 --> 00:11:00.280 Courant Events Right: Those systems are very interesting. 223 00:11:00.280 --> 00:11:01.530 Christophe Raymond Garban: They, they have a… 224 00:11:01.530 --> 00:11:05.790 Courant Events Right: More subtle phase transition, at least if the symmetry is a billion. 225 00:11:06.320 --> 00:11:07.479 Christophe Raymond Garban: And they noticed that. 226 00:11:07.480 --> 00:11:09.380 Courant Events Right: to… Atlanta. 227 00:11:09.380 --> 00:11:10.830 Christophe Raymond Garban: such temperature, the… 228 00:11:10.830 --> 00:11:12.230 Courant Events Right: coupon function, which measures. 229 00:11:12.230 --> 00:11:15.060 Christophe Raymond Garban: How the angle at that part of the. 230 00:11:15.060 --> 00:11:16.340 Courant Events Right: domain correlated. 231 00:11:16.340 --> 00:11:17.140 Christophe Raymond Garban: Thanks for dinner. 232 00:11:17.140 --> 00:11:17.800 Courant Events Right: at that point. 233 00:11:17.800 --> 00:11:18.999 Christophe Raymond Garban: part of the domain. 234 00:11:19.000 --> 00:11:21.049 Courant Events Right: They noticed that this must decay explod. 235 00:11:21.050 --> 00:11:22.900 Christophe Raymond Garban: Eventually, quickly, and this is quite. 236 00:11:22.900 --> 00:11:23.530 Courant Events Right: Easy to show. 237 00:11:23.530 --> 00:11:29.010 Christophe Raymond Garban: with two days' techniques when the temperature is large, so there's no surprise here. 238 00:11:29.010 --> 00:11:30.139 Courant Events Right: The surprise was that if the. 239 00:11:30.140 --> 00:11:32.380 Christophe Raymond Garban: Temperature is low, certainly the system. 240 00:11:32.380 --> 00:11:35.930 Courant Events Right: Enter the phase where the decay here. 241 00:11:36.120 --> 00:11:37.750 Christophe Raymond Garban: becomes collar. 242 00:11:37.940 --> 00:11:38.429 Christophe Raymond Garban: And that's. 243 00:11:38.430 --> 00:11:40.350 Courant Events Right: at least in this U-shaped. 244 00:11:40.350 --> 00:11:42.270 Christophe Raymond Garban: We should all agree that. 245 00:11:42.270 --> 00:11:43.089 Courant Events Right: Ovalu is now. 246 00:11:43.090 --> 00:11:44.490 Christophe Raymond Garban: Right? This is maybe. 247 00:11:44.490 --> 00:11:49.680 Courant Events Right: Maybe a sign of conformal field theory, of large-scale structures, of lots of good mathematics. 248 00:11:49.870 --> 00:11:51.930 Courant Events Right: And here, we have an antagonist. 249 00:11:51.930 --> 00:11:55.059 Christophe Raymond Garban: line of, of models, when I vary the time. 250 00:11:55.060 --> 00:11:56.529 Courant Events Right: temperature below the face foundation. 251 00:11:57.120 --> 00:12:02.830 Courant Events Right: And each of those will be associated to some critical exponents which encode this decay. 252 00:12:03.130 --> 00:12:05.930 Courant Events Right: If we go back to the easing model, we would have. 253 00:12:05.930 --> 00:12:09.059 Christophe Raymond Garban: One point, on the other point, maybe, of the. 254 00:12:09.060 --> 00:12:12.290 Courant Events Right: your oppon Only at that point, you would have power load decay. 255 00:12:12.290 --> 00:12:13.010 Christophe Raymond Garban: He is. 256 00:12:13.010 --> 00:12:17.679 Courant Events Right: It's very different. At that point, included and below, we have power low. 257 00:12:19.240 --> 00:12:20.530 Courant Events Right: So, what I want to do 258 00:12:20.850 --> 00:12:27.180 Courant Events Right: in the next few slides is to give you the physics way of understanding this phase transition. 259 00:12:27.350 --> 00:12:29.640 Courant Events Right: Which is beautiful, I find. 260 00:12:29.970 --> 00:12:34.249 Christophe Raymond Garban: And then I will give you the mass way. It's not orthogonal to the physics way, but I claim it. 261 00:12:34.250 --> 00:12:35.450 Courant Events Right: It's a little bit different. 262 00:12:36.050 --> 00:12:39.779 Courant Events Right: So, how did Berzinski, Kostoletz Staus. 263 00:12:39.780 --> 00:12:46.080 Christophe Raymond Garban: explained this phase from the They started with the Gibbs measure. 264 00:12:46.390 --> 00:12:48.950 Christophe Raymond Garban: which I had written before, so it was. 265 00:12:48.950 --> 00:12:52.530 Courant Events Right: financial beta Some over edges of my graph, which. 266 00:12:52.530 --> 00:12:54.289 Christophe Raymond Garban: So the nearest neighbor ages. 267 00:12:54.290 --> 00:12:55.450 Courant Events Right: Scala product. 268 00:12:55.450 --> 00:13:02.519 Christophe Raymond Garban: of the S1 spin at I with the S1 spin at the point just next to it, which is J. 269 00:13:02.950 --> 00:13:06.319 Christophe Raymond Garban: The scalar product of S1 value spins, if you. 270 00:13:06.320 --> 00:13:07.380 Courant Events Right: You know about. 271 00:13:07.380 --> 00:13:10.989 Christophe Raymond Garban: This is just a cosine of the difference of angles. 272 00:13:11.250 --> 00:13:14.829 Christophe Raymond Garban: So, a good idea is to reparametrize the S1 value. 273 00:13:14.830 --> 00:13:17.219 Courant Events Right: spin system by 0 to 2 power. 274 00:13:17.220 --> 00:13:19.629 Christophe Raymond Garban: angles theta i, where. 275 00:13:19.630 --> 00:13:21.240 Courant Events Right: Why are still the vertices. 276 00:13:21.240 --> 00:13:22.410 Christophe Raymond Garban: of Minecraft. 277 00:13:22.600 --> 00:13:25.040 Courant Events Right: So you can write down the Gibbs measure this way. 278 00:13:25.120 --> 00:13:29.409 Christophe Raymond Garban: And now you… Try to tailor-expand this cosine. 279 00:13:29.410 --> 00:13:35.420 Courant Events Right: With the idea that if beta is very large, maybe the gradient will tend to be typically very small. 280 00:13:35.880 --> 00:13:40.410 Courant Events Right: So if you do this, you, at large scales, you may understand your Gibbs. 281 00:13:40.410 --> 00:13:43.600 Christophe Raymond Garban: measure as exponential minus beta over 2. 282 00:13:43.710 --> 00:13:46.230 Courant Events Right: And the energy that you are analyzing. 283 00:13:46.230 --> 00:13:47.869 Christophe Raymond Garban: I think it's a Dirichlet Energy. 284 00:13:47.870 --> 00:13:48.749 Courant Events Right: on the field. 285 00:13:48.750 --> 00:13:49.900 Christophe Raymond Garban: of angers. 286 00:13:50.230 --> 00:13:51.119 Courant Events Right: And now comes May. 287 00:13:51.120 --> 00:13:53.430 Christophe Raymond Garban: Maybe the main, observation. 288 00:13:53.540 --> 00:13:59.589 Courant Events Right: This field of angles in this two-dimensional plane, you should not think of it as a. 289 00:13:59.590 --> 00:14:03.670 Christophe Raymond Garban: 0 form as a function, but instead as a one form. 290 00:14:03.910 --> 00:14:04.660 Christophe Raymond Garban: Why that? 291 00:14:04.660 --> 00:14:06.139 Courant Events Right: But, if I take a point here. 292 00:14:06.140 --> 00:14:07.430 Christophe Raymond Garban: And the point here? 293 00:14:07.430 --> 00:14:11.389 Courant Events Right: And now I take any pass which goes from this first point to the second point. 294 00:14:12.100 --> 00:14:18.289 Courant Events Right: If beta is very large, if the temperature is very small, the gradients of angles are going to be very, very tiny. 295 00:14:18.750 --> 00:14:19.280 Christophe Raymond Garban: So. 296 00:14:19.280 --> 00:14:20.570 Courant Events Right: It will be possible to make. 297 00:14:20.570 --> 00:14:23.649 Christophe Raymond Garban: So, how does the angle move along. 298 00:14:23.650 --> 00:14:26.739 Courant Events Right: this path? How do I move continuously along the S1. 299 00:14:26.740 --> 00:14:27.540 Christophe Raymond Garban: Let's see here. 300 00:14:27.540 --> 00:14:28.110 Courant Events Right: Okay. 301 00:14:28.650 --> 00:14:33.920 Christophe Raymond Garban: But if I were to take a completely different path here, it will still be very well defined, but it might. 302 00:14:33.920 --> 00:14:34.490 Courant Events Right: Maybe that when. 303 00:14:34.490 --> 00:14:36.389 Christophe Raymond Garban: And reach the same final point. 304 00:14:36.510 --> 00:14:43.300 Courant Events Right: The angle that I will have computed will be the same modulo and monodromy of plus 2 pi plus 4 pi minus 2 pi. 305 00:14:43.750 --> 00:14:44.970 Courant Events Right: So, all this to say. 306 00:14:45.670 --> 00:14:53.460 Courant Events Right: a path, I can associate a gradient of angle, but different paths may not agree, and they may accumulate monodromies in between. 307 00:14:54.990 --> 00:14:56.519 Courant Events Right: This meant that these. 308 00:14:56.520 --> 00:14:59.419 Christophe Raymond Garban: Random, one form that we're trying to understand. 309 00:14:59.520 --> 00:15:02.620 Christophe Raymond Garban: Its randomness may arise from two things. 310 00:15:02.620 --> 00:15:04.959 Courant Events Right: arise from a zero form, on the one hand. 311 00:15:05.190 --> 00:15:07.110 Christophe Raymond Garban: And it may also arise from a. 312 00:15:07.110 --> 00:15:11.040 Courant Events Right: Two forms if this one form happened to be non-exact. 313 00:15:12.350 --> 00:15:21.309 Courant Events Right: So, that's the basis of their analysis, and if the model is slightly different from the XY model, it's called the VIN model, I will not define it, but it's a very close cuisine. 314 00:15:21.880 --> 00:15:22.420 Christophe Raymond Garban: It's good. 315 00:15:22.420 --> 00:15:25.459 Courant Events Right: It can be rigorously proved that the zero form. 316 00:15:25.460 --> 00:15:25.950 Christophe Raymond Garban: So… 317 00:15:25.950 --> 00:15:29.529 Courant Events Right: of randomness is actually independent from the two-form source of. 318 00:15:29.530 --> 00:15:29.880 Christophe Raymond Garban: random. 319 00:15:30.780 --> 00:15:32.039 Courant Events Right: This one is known as. 320 00:15:32.040 --> 00:15:35.239 Christophe Raymond Garban: Gaussian-free field, and this gives rise to the. 321 00:15:35.240 --> 00:15:37.830 Courant Events Right: complex, exponential. 322 00:15:37.830 --> 00:15:39.649 Christophe Raymond Garban: GFF, which is a very nice. 323 00:15:39.650 --> 00:15:40.190 Courant Events Right: objective. 324 00:15:40.190 --> 00:15:42.919 Christophe Raymond Garban: its own, as we've seen in previous talks. 325 00:15:43.050 --> 00:15:48.450 Courant Events Right: But if you take the complex exponential of a GFF, you will never see any BKD transition. 326 00:15:48.670 --> 00:15:56.419 Courant Events Right: The only mechanism to break power law into exponential low will come from the coolant gas, which I will not define. 327 00:15:56.420 --> 00:15:56.929 Christophe Raymond Garban: I haven't took. 328 00:15:56.930 --> 00:15:58.000 Courant Events Right: about in the next slide. 329 00:15:59.220 --> 00:16:00.459 Christophe Raymond Garban: So, what is… 330 00:16:00.460 --> 00:16:01.220 Courant Events Right: this cooling gaps. 331 00:16:01.220 --> 00:16:01.800 Christophe Raymond Garban: F? 332 00:16:02.060 --> 00:16:06.050 Christophe Raymond Garban: The source of randomness coming from the two forms describes. 333 00:16:06.050 --> 00:16:10.669 Courant Events Right: Places where we do accumulate monodromies around these plackets. 334 00:16:11.110 --> 00:16:15.070 Courant Events Right: And the argument of Berzinski, Kosolitz, and Taurus is that every time. 335 00:16:15.070 --> 00:16:15.990 Christophe Raymond Garban: you have. 336 00:16:16.330 --> 00:16:20.040 Courant Events Right: Monodromy, such a defect that will do topological defects. 337 00:16:20.150 --> 00:16:30.559 Courant Events Right: You will need to find somewhere else an anti-vortex, so that if you take the integral around the big box, you don't accumulate too much angles around the box. 338 00:16:30.750 --> 00:16:32.910 Courant Events Right: So typically, a vortex will come. 339 00:16:32.910 --> 00:16:34.000 Christophe Raymond Garban: With an anti-vortex. 340 00:16:34.210 --> 00:16:35.940 Courant Events Right: At a certain distance. 341 00:16:36.770 --> 00:16:37.770 Courant Events Right: And now. 342 00:16:37.770 --> 00:16:40.730 Christophe Raymond Garban: a similar argument as I did for the. 343 00:16:40.730 --> 00:16:45.040 Courant Events Right: easy model. It was targeted to do it now for the XY model. 344 00:16:45.350 --> 00:16:47.379 Courant Events Right: You can ask yourself what's. 345 00:16:47.380 --> 00:16:49.010 Christophe Raymond Garban: It's the energy that you. 346 00:16:49.010 --> 00:16:53.760 Courant Events Right: Pay if you want to produce an isolated pair of vortex and anti-vortex. 347 00:16:54.120 --> 00:16:57.450 Courant Events Right: It's easy to check that if you try to compute the sum. 348 00:16:57.450 --> 00:16:58.339 Christophe Raymond Garban: over the edge. 349 00:16:58.340 --> 00:17:00.140 Courant Events Right: of the gradient squared here. 350 00:17:00.580 --> 00:17:01.280 Christophe Raymond Garban: What you will… 351 00:17:01.280 --> 00:17:02.350 Courant Events Right: And the quiz isn't. 352 00:17:02.350 --> 00:17:04.890 Christophe Raymond Garban: constant times the log of the Euclidean. 353 00:17:04.890 --> 00:17:07.920 Courant Events Right: The distance between the vortex and the antivortex. 354 00:17:08.220 --> 00:17:15.900 Courant Events Right: So each isolated pair of vortex and antivortex will come with a log n energy price. 355 00:17:16.060 --> 00:17:19.929 Courant Events Right: In such a way that if beta is small, if the temperature is high. 356 00:17:20.069 --> 00:17:21.740 Christophe Raymond Garban: Every pair of vortex will. 357 00:17:21.740 --> 00:17:23.740 Courant Events Right: Pay this polynomial cost. 358 00:17:23.900 --> 00:17:24.520 Courant Events Right: But the… 359 00:17:24.520 --> 00:17:25.060 Christophe Raymond Garban: attribute. 360 00:17:25.060 --> 00:17:34.260 Courant Events Right: way of choosing a vortex and an antivortex is n to the 4, basically, so the entropy at large temperature may easily. 361 00:17:34.260 --> 00:17:34.880 Christophe Raymond Garban: in over. 362 00:17:34.880 --> 00:17:35.800 Courant Events Right: the energy. 363 00:17:36.660 --> 00:17:37.180 Courant Events Right: So… 364 00:17:37.180 --> 00:17:37.640 Christophe Raymond Garban: No. 365 00:17:37.640 --> 00:17:48.540 Courant Events Right: As beta grows, at some point, this energy term will start to be too much compared to the entropy, and that's basically where the BKT transition will happen. 366 00:17:49.730 --> 00:17:52.369 Courant Events Right: So there is an analog situation, it's called. 367 00:17:52.370 --> 00:17:53.670 Christophe Raymond Garban: two components. 368 00:17:53.670 --> 00:17:55.090 Courant Events Right: coolant gas in R2. 369 00:17:55.320 --> 00:17:56.129 Christophe Raymond Garban: And there's a… 370 00:17:56.130 --> 00:17:57.880 Courant Events Right: Very nice recent work. 371 00:17:57.880 --> 00:17:58.859 Christophe Raymond Garban: Jean Bourussier. 372 00:17:58.860 --> 00:17:59.790 Courant Events Right: Sylvia Serati. 373 00:18:00.020 --> 00:18:00.639 Christophe Raymond Garban: Where the… 374 00:18:00.640 --> 00:18:01.719 Courant Events Right: analyze how the. 375 00:18:01.720 --> 00:18:02.669 Christophe Raymond Garban: is, voice. 376 00:18:02.670 --> 00:18:04.889 Courant Events Right: Vortices and antivortices cluster together. 377 00:18:04.890 --> 00:18:07.520 Christophe Raymond Garban: all the way to the BKT transition. 378 00:18:08.830 --> 00:18:09.520 Christophe Raymond Garban: So back… 379 00:18:09.520 --> 00:18:10.650 Courant Events Right: to the lattice? 380 00:18:11.030 --> 00:18:11.490 Christophe Raymond Garban: We've beaten. 381 00:18:11.490 --> 00:18:12.880 Courant Events Right: is now very large. 382 00:18:12.880 --> 00:18:13.440 Christophe Raymond Garban: Nope. 383 00:18:13.480 --> 00:18:18.020 Courant Events Right: What happens is that you will still have vortices and antivortices everywhere. 384 00:18:18.020 --> 00:18:18.420 Christophe Raymond Garban: revenue. 385 00:18:18.420 --> 00:18:29.369 Courant Events Right: Huti domain But it would be very hard for them to be separated from each other at macroscopic distance. 386 00:18:31.010 --> 00:18:37.519 Courant Events Right: Yes, the easy computation here is for one isolated pair. If you start having a typical configuration, which is a 387 00:18:38.000 --> 00:18:42.650 Courant Events Right: positive density of vortices, it's much harder to make sense of this. 388 00:18:43.380 --> 00:18:45.040 Courant Events Right: And that's why I… 389 00:18:45.040 --> 00:18:45.760 Christophe Raymond Garban: Right. 390 00:18:45.760 --> 00:18:49.670 Courant Events Right: It's hard to make these, heuristics by Berzinski Cosarest. 391 00:18:49.860 --> 00:18:50.660 Christophe Raymond Garban: a rigorous. 392 00:18:51.470 --> 00:18:55.049 Courant Events Right: Okay. But if you just had one pair of vortices on two vertices, then. 393 00:18:55.360 --> 00:19:01.609 Christophe Raymond Garban: You can compute the optimal way of achieving this, it's like a harmonic map, and this costs log N. 394 00:19:02.260 --> 00:19:04.480 Courant Events Right: And so this argument is for comparing 395 00:19:05.210 --> 00:19:09.400 Courant Events Right: Very sparse collection of pairs with having nothing at all. 396 00:19:10.000 --> 00:19:14.250 Christophe Raymond Garban: Exactly. So, maybe, compared to the easing model, I should say that. 397 00:19:14.340 --> 00:19:20.010 Courant Events Right: The XY model has this GFF fluctuation, which… which will. 398 00:19:20.010 --> 00:19:24.640 Christophe Raymond Garban: always be power low. And on top of this GFF, you have these vortices which will. 399 00:19:24.640 --> 00:19:26.000 Courant Events Right: even more disorder. 400 00:19:26.400 --> 00:19:28.590 Christophe Raymond Garban: And when you have tons of those, it far only. 401 00:19:28.590 --> 00:19:31.360 Courant Events Right: from each other, it actually produces exponential decay. 402 00:19:32.450 --> 00:19:34.610 Courant Events Right: So this transition is also called the vertical. 403 00:19:34.610 --> 00:19:35.070 Christophe Raymond Garban: on. 404 00:19:35.070 --> 00:19:37.719 Courant Events Right: Vortex binding, unbinding phenomenon. 405 00:19:38.380 --> 00:19:40.690 Christophe Raymond Garban: And that's the physics way. 406 00:19:40.820 --> 00:19:43.789 Courant Events Right: Unless there are questions, I want you to go to the mathematical. 407 00:19:43.790 --> 00:19:44.400 Christophe Raymond Garban: right now. 408 00:19:45.980 --> 00:19:47.040 Christophe Raymond Garban: And this is the thing. 409 00:19:47.040 --> 00:19:50.889 Courant Events Right: also stolen from the web managers, tried to indicate the vortices. 410 00:19:51.140 --> 00:19:52.080 Christophe Raymond Garban: Versus the online. 411 00:19:52.080 --> 00:19:53.090 Courant Events Right: offices, hopefully. 412 00:19:53.090 --> 00:19:54.610 Christophe Raymond Garban: There's nothing communicate? 413 00:19:57.420 --> 00:19:59.620 Courant Events Right: So, I will move now to the way. 414 00:19:59.790 --> 00:20:00.580 Christophe Raymond Garban: Matthew, mate. 415 00:20:00.580 --> 00:20:03.460 Courant Events Right: Politicians have managed to… Inc. 416 00:20:03.460 --> 00:20:08.460 Christophe Raymond Garban: prove these, predictions from BK teams? Yep. 417 00:20:09.630 --> 00:20:13.099 Courant Events Right: There is some… can you come up with some artificial model. 418 00:20:13.990 --> 00:20:17.910 Christophe Raymond Garban: Where you could tune the progress into vortices so that. 419 00:20:17.910 --> 00:20:18.809 Courant Events Right: To be clear that. 420 00:20:19.390 --> 00:20:20.749 Christophe Raymond Garban: The signal bear domain. 421 00:20:20.750 --> 00:20:21.569 Courant Events Right: It's a very pillow. 422 00:20:21.570 --> 00:20:22.520 Christophe Raymond Garban: facility. 423 00:20:22.520 --> 00:20:23.360 Courant Events Right: religion. 424 00:20:23.770 --> 00:20:24.910 Christophe Raymond Garban: I wore the cool guys. 425 00:20:25.180 --> 00:20:29.790 Christophe Raymond Garban: So, for the particular latest model you are considering, of course, you don't have the. 426 00:20:29.790 --> 00:20:30.360 Courant Events Right: motion. 427 00:20:30.840 --> 00:20:31.859 Courant Events Right: You guys can do speed. 428 00:20:31.860 --> 00:20:34.579 Christophe Raymond Garban: So I don't know if this answers. 429 00:20:34.580 --> 00:20:48.420 Courant Events Right: the question, but you can, on the lattice, you can make a Coulomb gas which has a Z to the power, the number of… and if you do this, then there is a very nice diagram which captures Z and beta simultaneously. 430 00:20:48.430 --> 00:20:49.479 Christophe Raymond Garban: And along this. 431 00:20:49.480 --> 00:20:51.230 Courant Events Right: diagram, you can see the RG flow. 432 00:20:51.690 --> 00:20:52.280 Christophe Raymond Garban: And… 433 00:20:52.280 --> 00:20:54.419 Courant Events Right: There is a very nice work. 434 00:20:54.420 --> 00:20:55.690 Christophe Raymond Garban: I wouldn't, I wouldn't. 435 00:20:55.690 --> 00:21:00.299 Courant Events Right: state this work in 5 minutes, where in some sense, they use this 436 00:21:00.640 --> 00:21:10.240 Courant Events Right: small fugacity explanation. And where are you in better shape with respect to this experiment? So, as far as I understand, if your fugacity is extremely small. 437 00:21:10.240 --> 00:21:10.790 Christophe Raymond Garban: then you. 438 00:21:10.790 --> 00:21:11.210 Courant Events Right: clean them. 439 00:21:11.590 --> 00:21:13.889 Courant Events Right: shape, you can actually even compute exponents. 440 00:21:14.650 --> 00:21:17.530 Christophe Raymond Garban: So there is a… Pierre Lucie Falcobe. 441 00:21:17.530 --> 00:21:19.220 Courant Events Right: There is also a work I will mention. 442 00:21:19.220 --> 00:21:20.500 Christophe Raymond Garban: of. 443 00:21:20.500 --> 00:21:22.500 Courant Events Right: Borisin Park in Rodriguez Apottis. 444 00:21:24.520 --> 00:21:26.140 Courant Events Right: Could you repeat the questions? 445 00:21:26.140 --> 00:21:26.840 Christophe Raymond Garban: It's hard to… 446 00:21:26.840 --> 00:21:28.360 Courant Events Right: Oh, yeah, so the question was. 447 00:21:28.360 --> 00:21:31.929 Christophe Raymond Garban: here, I will rephrase it here. If I look at. 448 00:21:31.930 --> 00:21:35.270 Courant Events Right: the Villan model, or the XY model, 449 00:21:35.780 --> 00:21:36.470 Christophe Raymond Garban: They are what. 450 00:21:36.470 --> 00:21:39.159 Courant Events Right: Vortices and antivortices all over the place. 451 00:21:40.170 --> 00:21:51.489 Courant Events Right: try to generalize this model by adding augacity which maybe forces more vortices or less vortices, so it's another parameter which does not exist in this model. 452 00:21:51.650 --> 00:21:56.879 Courant Events Right: And if you play with this fugacity, when the fugacity is small, you can reach more phenom energy than. 453 00:21:56.980 --> 00:21:57.920 Christophe Raymond Garban: In these material elections. 454 00:21:57.920 --> 00:21:59.109 Courant Events Right: more exponents. 455 00:21:59.410 --> 00:22:02.159 Courant Events Right: It's additional modified the line. 456 00:22:02.160 --> 00:22:07.770 Christophe Raymond Garban: The field which is part of the plaquettes, you treat as a background field, I mean. 457 00:22:08.920 --> 00:22:17.649 Courant Events Right: Right, so, so I think, I will try to rephrase. So, is it a general… the question is, is it, like, in the general. 458 00:22:17.650 --> 00:22:18.000 Christophe Raymond Garban: horizon. 459 00:22:18.000 --> 00:22:18.430 Courant Events Right: Pilot. 460 00:22:18.870 --> 00:22:19.679 Christophe Raymond Garban: So I think… 461 00:22:19.680 --> 00:22:25.350 Courant Events Right: Indeed, the VLAN model is something you could see as a model which leads along the edges of the graph. 462 00:22:25.950 --> 00:22:33.240 Courant Events Right: And you could ask for each placket, you could penalize that it's actually trying to make a monodromy there. And then, yes, indeed, it would be a bit like. 463 00:22:33.430 --> 00:22:34.650 Christophe Raymond Garban: having a small figure. 464 00:22:34.650 --> 00:22:35.859 Courant Events Right: CTZ. That's correct. 465 00:22:39.680 --> 00:22:40.590 Christophe Raymond Garban: Okay, so in… 466 00:22:40.590 --> 00:22:46.190 Courant Events Right: in, in Nikola's, Right. 467 00:22:46.190 --> 00:22:47.700 Christophe Raymond Garban: Very balanced so far. 468 00:22:47.700 --> 00:22:48.440 Courant Events Right: Nope. 469 00:22:49.540 --> 00:22:50.420 Christophe Raymond Garban: Okay, so… 470 00:22:50.420 --> 00:22:51.589 Courant Events Right: A few words about the… 471 00:22:51.590 --> 00:22:54.279 Christophe Raymond Garban: The way it has been understood. 472 00:22:54.280 --> 00:22:55.380 Courant Events Right: in math, so the. 473 00:22:55.380 --> 00:22:56.430 Christophe Raymond Garban: the brick… 474 00:22:56.430 --> 00:22:57.709 Courant Events Right: Through work here was. 475 00:22:57.710 --> 00:22:58.100 Christophe Raymond Garban: in… 476 00:22:58.100 --> 00:23:00.540 Courant Events Right: in the early 80s by Frolis and Spencer. 477 00:23:01.060 --> 00:23:04.529 Christophe Raymond Garban: And, to make the story short. 478 00:23:04.530 --> 00:23:10.740 Courant Events Right: I could maybe say the following. If you want to prove that a phase transition holds for a S1 value. 479 00:23:10.740 --> 00:23:12.210 Christophe Raymond Garban: a good model, which… 480 00:23:12.210 --> 00:23:16.319 Courant Events Right: Goes from exponential to quasi-long-range order. 481 00:23:16.320 --> 00:23:19.279 Christophe Raymond Garban: There is a difficulty, which is that. 482 00:23:19.280 --> 00:23:21.279 Courant Events Right: You don't have any order parameters. 483 00:23:21.280 --> 00:23:22.969 Christophe Raymond Garban: to play with a pre-ring. 484 00:23:23.660 --> 00:23:24.240 Courant Events Right: the idea. 485 00:23:24.240 --> 00:23:24.640 Christophe Raymond Garban: That's too… 486 00:23:24.640 --> 00:23:25.650 Courant Events Right: Go to a different. 487 00:23:25.650 --> 00:23:30.549 Christophe Raymond Garban: where the order parameter was easier to find, and what they've done is that. 488 00:23:30.550 --> 00:23:31.680 Courant Events Right: They used that the four-year. 489 00:23:31.680 --> 00:23:37.960 Christophe Raymond Garban: dual of S1 is Z. So they basically wrote this model in the Fourier, 490 00:23:37.960 --> 00:23:48.750 Courant Events Right: side, so they Fourier transform the XY or the VIN model, and they end up with a dual model, which is now living on the plackets of Z2, but the. 491 00:23:48.820 --> 00:23:51.470 Christophe Raymond Garban: C is equal. 492 00:23:51.650 --> 00:23:52.429 Christophe Raymond Garban: And each… 493 00:23:52.430 --> 00:23:56.999 Courant Events Right: Each such placket will now carry not a S1-valued spin, but a Z-valued spin. 494 00:23:57.640 --> 00:23:58.690 Christophe Raymond Garban: So this is called a… 495 00:23:58.690 --> 00:24:00.360 Courant Events Right: Z valued height function. 496 00:24:00.890 --> 00:24:09.060 Courant Events Right: And the easy part of BKT, at least on the lattice, is sent to the easy part of the dual model, which is a low temperature. 497 00:24:09.260 --> 00:24:11.760 Christophe Raymond Garban: Z-valued, head function. 498 00:24:11.760 --> 00:24:12.550 Courant Events Right: model, which. 499 00:24:12.670 --> 00:24:13.480 Christophe Raymond Garban: Locally… 500 00:24:13.480 --> 00:24:14.449 Courant Events Right: At least at. 501 00:24:14.450 --> 00:24:17.990 Christophe Raymond Garban: Very high temperature, yeah, very low, the dual temperature is the inverse. 502 00:24:18.470 --> 00:24:24.189 Christophe Raymond Garban: Payroll's argument comes back, excitations, as I did before, comes back. 503 00:24:24.190 --> 00:24:26.440 Courant Events Right: The model will typically be very flat. 504 00:24:27.030 --> 00:24:28.710 Courant Events Right: The difficult part was here. 505 00:24:28.710 --> 00:24:30.220 Christophe Raymond Garban: Which is incorrect. 506 00:24:30.220 --> 00:24:33.449 Courant Events Right: with the difficult part of BKT, and there… 507 00:24:33.450 --> 00:24:35.580 Christophe Raymond Garban: They were able to prove that. 508 00:24:35.580 --> 00:24:42.229 Courant Events Right: the Z-valued Gaussian-free field, or the dual of the XY model, which is a bit more complicated to write down. 509 00:24:42.440 --> 00:24:42.920 Christophe Raymond Garban: They look… 510 00:24:42.920 --> 00:24:44.350 Courant Events Right: localizes at, 511 00:24:44.350 --> 00:24:49.369 Christophe Raymond Garban: low temperature for XY, which is close to high temperature for the dual model. 512 00:24:49.680 --> 00:24:52.490 Christophe Raymond Garban: This delocalization property was the hard part. 513 00:24:52.490 --> 00:24:53.340 Courant Events Right: Oof! 514 00:24:53.840 --> 00:24:54.820 Courant Events Right: And, 515 00:24:54.990 --> 00:24:56.020 Christophe Raymond Garban: In fact. 516 00:24:56.020 --> 00:24:59.450 Courant Events Right: The way that the proof uses a little bit intuition. 517 00:24:59.450 --> 00:25:03.950 Christophe Raymond Garban: the Coulomb gas, a little bit of intuition from binding on binding, but not. 518 00:25:03.950 --> 00:25:04.370 Courant Events Right: Exactly. 519 00:25:04.370 --> 00:25:04.980 Christophe Raymond Garban: bet. 520 00:25:05.450 --> 00:25:07.629 Courant Events Right: Is this C-valid height function same? 521 00:25:07.630 --> 00:25:13.059 Christophe Raymond Garban: So the Z value, the height function, it does. 522 00:25:13.060 --> 00:25:13.929 Courant Events Right: It doesn't have monotone. 523 00:25:13.930 --> 00:25:18.999 Christophe Raymond Garban: It's a really Oh, good question. It… so… 524 00:25:19.000 --> 00:25:20.190 Courant Events Right: If I take a VLAN. 525 00:25:20.190 --> 00:25:22.459 Christophe Raymond Garban: and I look at the fields of. 526 00:25:22.460 --> 00:25:27.979 Courant Events Right: monodrome is. This is a very nice poeticistic object, it's exactly a cooling gas, so it's a. 527 00:25:27.980 --> 00:25:30.790 Christophe Raymond Garban: integer-valued Gaussian. 528 00:25:31.060 --> 00:25:34.550 Courant Events Right: This model is exactly the information to. 529 00:25:34.550 --> 00:25:38.590 Christophe Raymond Garban: early, so… Not a monotopy, but very related. 530 00:25:40.440 --> 00:25:42.810 Christophe Raymond Garban: Isn't it true that in this… 531 00:25:43.260 --> 00:25:48.210 Christophe Raymond Garban: mathematics Pro, there's also an interesting The education group under. 532 00:25:48.210 --> 00:25:51.419 Courant Events Right: interpretation on the hard part, where you're trying to. 533 00:25:51.420 --> 00:25:52.480 Christophe Raymond Garban: to start. 534 00:25:53.610 --> 00:25:54.090 Christophe Raymond Garban: You don't. 535 00:25:54.090 --> 00:25:54.430 Courant Events Right: won't. 536 00:25:54.430 --> 00:26:01.490 Christophe Raymond Garban: which has a uniform Distribution fields, and then it is determined by some periodic potential. 537 00:26:01.770 --> 00:26:03.350 Christophe Raymond Garban: Right, so I think these… 538 00:26:03.350 --> 00:26:04.580 Courant Events Right: hard refloat. 539 00:26:04.870 --> 00:26:06.370 Courant Events Right: That you are mentioning is exactly. 540 00:26:06.370 --> 00:26:12.020 Christophe Raymond Garban: I will mention it in this slide. 541 00:26:14.240 --> 00:26:15.159 Courant Events Right: I think visiting. 542 00:26:15.160 --> 00:26:21.289 Christophe Raymond Garban: Not even that physics is not… physics, people think in terms of both of them, especially nowadays. 543 00:26:23.370 --> 00:26:25.810 Christophe Raymond Garban: Okay, make me know what it is. 544 00:26:25.810 --> 00:26:28.620 Courant Events Right: I read Blazinski Costco Restore as the compute login. 545 00:26:28.620 --> 00:26:29.930 Christophe Raymond Garban: Again, we say expansion. 546 00:26:29.930 --> 00:26:30.840 Courant Events Right: tremendous detail. 547 00:26:30.840 --> 00:26:32.610 Christophe Raymond Garban: It didn't find until… 548 00:26:36.160 --> 00:26:43.350 Christophe Raymond Garban: But, okay, I think what you have in mind, maybe I will switch to this one. If you take this dual model. 549 00:26:43.350 --> 00:26:43.910 Courant Events Right: measure. 550 00:26:44.740 --> 00:26:46.340 Christophe Raymond Garban: So, beta is large. 551 00:26:46.340 --> 00:26:47.780 Courant Events Right: is… this is a… 552 00:26:47.780 --> 00:26:50.260 Christophe Raymond Garban: I inverted the usual temperature. 553 00:26:50.780 --> 00:26:51.500 Christophe Raymond Garban: So, in… 554 00:26:51.500 --> 00:26:52.090 Courant Events Right: In this case. 555 00:26:52.090 --> 00:26:53.100 Christophe Raymond Garban: case, 556 00:26:53.100 --> 00:26:55.309 Courant Events Right: If you are at a… Very large beta. 557 00:26:55.310 --> 00:26:56.170 Christophe Raymond Garban: Yeah, which is, like. 558 00:26:56.170 --> 00:26:58.660 Courant Events Right: very high temperature Z-valued GFF. 559 00:26:59.160 --> 00:27:03.480 Christophe Raymond Garban: Bojmines Paprodeguise, in a very important work, have shown that. 560 00:27:03.600 --> 00:27:04.509 Courant Events Right: At large scale. 561 00:27:04.510 --> 00:27:06.939 Christophe Raymond Garban: This will look like a GFF. 562 00:27:07.120 --> 00:27:08.970 Christophe Raymond Garban: But the temperature may be… 563 00:27:08.970 --> 00:27:09.500 Courant Events Right: that it's. 564 00:27:09.500 --> 00:27:10.130 Christophe Raymond Garban: Please. 565 00:27:11.020 --> 00:27:11.550 Christophe Raymond Garban: And… 566 00:27:11.550 --> 00:27:12.469 Courant Events Right: Here, indeed, it's. 567 00:27:12.470 --> 00:27:13.350 Christophe Raymond Garban: So, in. 568 00:27:13.350 --> 00:27:14.619 Courant Events Right: in the realm of sign. 569 00:27:14.620 --> 00:27:18.379 Christophe Raymond Garban: And there is an activity, and you run it slow. 570 00:27:18.380 --> 00:27:23.469 Courant Events Right: And this also follows a physics argument, I fully agree with this, but the 70s, I'm not. 571 00:27:24.710 --> 00:27:26.230 Christophe Raymond Garban: As sure as that. 572 00:27:27.210 --> 00:27:28.090 Christophe Raymond Garban: So there's… 573 00:27:28.090 --> 00:27:29.210 Courant Events Right: risky work? 574 00:27:29.660 --> 00:27:30.370 Courant Events Right: Proof of the… 575 00:27:30.370 --> 00:27:34.700 Christophe Raymond Garban: Here, as I mentioned, was by Fish and Spencer. There is a… 576 00:27:34.910 --> 00:27:35.300 Courant Events Right: And more… 577 00:27:35.300 --> 00:27:36.239 Christophe Raymond Garban: taught by people. 578 00:27:36.240 --> 00:27:39.669 Courant Events Right: So, his work is based more on the techniques developed by. 579 00:27:39.670 --> 00:27:41.279 Christophe Raymond Garban: Good Sheffield. 580 00:27:41.830 --> 00:27:48.129 Christophe Raymond Garban: Based on this new proof by Pete Lovers, there are now new proofs of the BKTS condition. All of these proofs booked. 581 00:27:48.130 --> 00:27:48.670 Courant Events Right: through the. 582 00:27:48.670 --> 00:27:49.890 Christophe Raymond Garban: roulette compound? 583 00:27:50.030 --> 00:27:58.670 Christophe Raymond Garban: by von England Poglis and Eisenman, Ariel, and Shapiro, both works, give a very nice new point of view. 584 00:27:59.790 --> 00:28:01.700 Christophe Raymond Garban: More modest, but with… 585 00:28:01.700 --> 00:28:03.990 Courant Events Right: We could show that. 586 00:28:04.380 --> 00:28:15.519 Christophe Raymond Garban: If you look at this effective temperature, the hard part is the existence of it. We could show that it must deviate from what the GFF would do, so that's the contribution of the vortices. 587 00:28:15.990 --> 00:28:16.580 Christophe Raymond Garban: And… 588 00:28:17.150 --> 00:28:18.259 Christophe Raymond Garban: Last item, maybe. 589 00:28:18.260 --> 00:28:18.690 Courant Events Right: You don't. 590 00:28:18.690 --> 00:28:21.670 Christophe Raymond Garban: many directions. Last year, Earth and. 591 00:28:21.670 --> 00:28:25.050 Courant Events Right: and Schweiger, they could extend the delocalization to non-communic. 592 00:28:25.050 --> 00:28:26.350 Christophe Raymond Garban: insurance. 593 00:28:26.710 --> 00:28:28.320 Courant Events Right: And, today on Africa, you would have. 594 00:28:28.320 --> 00:28:29.940 Christophe Raymond Garban: A different proof of this. 595 00:28:30.780 --> 00:28:31.710 Courant Events Right: per W. 596 00:28:31.860 --> 00:28:32.930 Christophe Raymond Garban: And delivering them. 597 00:28:32.930 --> 00:28:33.530 Courant Events Right: Angular. 598 00:28:34.330 --> 00:28:38.310 Christophe Raymond Garban: Maybe one interesting aspect of this is that if you take the two. 599 00:28:38.310 --> 00:28:38.750 Courant Events Right: mad. 600 00:28:38.750 --> 00:28:46.000 Christophe Raymond Garban: of these. There are more general S1-valued models. They include the Villan, EXY, and many others. 601 00:28:46.130 --> 00:28:56.420 Christophe Raymond Garban: And one consequence of this paper is that all of you also have a BKT transformation, so it extends the possible models which you have PK. 602 00:28:58.500 --> 00:29:01.769 Courant Events Right: A last sentence about BKT, which. 603 00:29:01.770 --> 00:29:04.410 Christophe Raymond Garban: There were some, talks of. 604 00:29:04.410 --> 00:29:04.870 Courant Events Right: followed by. 605 00:29:04.870 --> 00:29:09.140 Christophe Raymond Garban: what's going on in hyperbolic geometry. So, I… 606 00:29:09.140 --> 00:29:09.830 Courant Events Right: I have a… 607 00:29:09.830 --> 00:29:11.170 Christophe Raymond Garban: A work in progress with. 608 00:29:11.170 --> 00:29:11.780 Courant Events Right: Do you think? 609 00:29:11.780 --> 00:29:19.110 Christophe Raymond Garban: So we identify a BK transition on the H2, which exists on H3. 610 00:29:19.360 --> 00:29:21.080 Christophe Raymond Garban: And just to explain. 611 00:29:21.080 --> 00:29:24.620 Courant Events Right: in one sitting where this comes from, there is the same battle. 612 00:29:24.620 --> 00:29:25.720 Christophe Raymond Garban: energy. 613 00:29:25.720 --> 00:29:26.680 Courant Events Right: entropy, then we have. 614 00:29:26.680 --> 00:29:27.820 Christophe Raymond Garban: invaluable. 615 00:29:28.530 --> 00:29:31.010 Courant Events Right: Why that? If I were to do an XY model. 616 00:29:31.010 --> 00:29:34.919 Christophe Raymond Garban: along this hyperbolic plane. Maybe I will also put. 617 00:29:34.920 --> 00:29:35.430 Courant Events Right: Spotify. 618 00:29:35.430 --> 00:29:40.750 Christophe Raymond Garban: vortices and antivortices, and as I did before, it's not very hard to. 619 00:29:40.750 --> 00:29:42.489 Courant Events Right: See that if I have a plus and a minus. 620 00:29:42.800 --> 00:29:43.680 Christophe Raymond Garban: I don't… 621 00:29:43.680 --> 00:29:45.250 Courant Events Right: symbolic distance and from its. 622 00:29:45.250 --> 00:29:55.219 Christophe Raymond Garban: This will cause a little bit of energy and distance. Linar in the hyperbolic distance, while in two dimensions can cause logic in the distance. 623 00:29:55.580 --> 00:30:00.759 Christophe Raymond Garban: But now, the battle is as in dimension 2 at exponential minus beta times. 624 00:30:00.760 --> 00:30:01.330 Courant Events Right: by February. 625 00:30:01.330 --> 00:30:02.270 Christophe Raymond Garban: Jasons? 626 00:30:02.300 --> 00:30:06.369 Courant Events Right: But the volume in hyperbolic graph itself is also exponential. 627 00:30:06.590 --> 00:30:09.039 Courant Events Right: So when you look at it, you feel that depending. 628 00:30:09.040 --> 00:30:10.630 Christophe Raymond Garban: On the value of beta, you may have. 629 00:30:10.630 --> 00:30:11.960 Courant Events Right: either go pick binding. 630 00:30:12.140 --> 00:30:16.980 Christophe Raymond Garban: All of this combined, and that's what we do in this workplace. 631 00:30:19.180 --> 00:30:21.900 Christophe Raymond Garban: Shouldn't just follow from following barriers for the Facebook. 632 00:30:24.130 --> 00:30:26.569 Courant Events Right: But the first transaction here is performing better in both. 633 00:30:26.570 --> 00:30:29.420 Christophe Raymond Garban: distinguishing. 634 00:30:29.530 --> 00:30:30.430 Christophe Raymond Garban: Blessed. 635 00:30:31.990 --> 00:30:34.240 Christophe Raymond Garban: I think… 636 00:30:34.240 --> 00:30:36.389 Courant Events Right: Here, it's a bit different, because the S1 637 00:30:37.590 --> 00:30:40.440 Courant Events Right: very different face transition. It will have, actually. 638 00:30:40.440 --> 00:30:41.050 Christophe Raymond Garban: intercom. 639 00:30:41.050 --> 00:30:45.230 Courant Events Right: No, no, H3 is the basic place. 640 00:30:49.150 --> 00:30:49.950 Christophe Raymond Garban: I mean. 641 00:30:49.950 --> 00:30:50.950 Courant Events Right: is from the S1. 642 00:30:50.950 --> 00:30:52.520 Christophe Raymond Garban: unvalued system. 643 00:30:52.520 --> 00:30:53.030 Courant Events Right: This is good. 644 00:30:53.030 --> 00:30:53.470 Christophe Raymond Garban: Thank you. 645 00:30:54.010 --> 00:30:59.300 Christophe Raymond Garban: a long range of face. Very different from the poisoning other. 646 00:30:59.300 --> 00:31:00.010 Courant Events Right: Unituent. 647 00:31:00.780 --> 00:31:08.259 Courant Events Right: So, we can discuss later, but I think it's quite different with energy. But the entropy energy, I agree, is fairly simple. 648 00:31:10.810 --> 00:31:11.710 Christophe Raymond Garban: Yep. 649 00:31:26.150 --> 00:31:28.710 Courant Events Right: Okay, that's a good point. 650 00:31:29.920 --> 00:31:31.089 Christophe Raymond Garban: Maybe I can… 651 00:31:31.090 --> 00:31:34.980 Courant Events Right: transfer, saying that we first tried to do the Frawley-Spencer argument. 652 00:31:36.120 --> 00:31:36.900 Christophe Raymond Garban: But… 653 00:31:36.900 --> 00:31:39.350 Courant Events Right: Push it further to try to have the. 654 00:31:39.350 --> 00:31:41.200 Christophe Raymond Garban: Binding, binding there. 655 00:31:41.340 --> 00:31:44.290 Christophe Raymond Garban: But because of the volume growing this fast. 656 00:31:44.290 --> 00:31:44.810 Courant Events Right: Combinator. 657 00:31:44.810 --> 00:31:49.269 Christophe Raymond Garban: it's collapsed. So, at least we are not a. 658 00:31:49.270 --> 00:31:51.069 Courant Events Right: But we did face an issue here. 659 00:31:51.400 --> 00:31:52.179 Christophe Raymond Garban: I'm sorry. 660 00:31:52.180 --> 00:31:52.760 Courant Events Right: Anything else? 661 00:31:55.920 --> 00:31:58.000 Christophe Raymond Garban: Yes? 662 00:31:58.290 --> 00:31:58.980 Christophe Raymond Garban: What might be? 663 00:31:59.310 --> 00:32:00.429 Christophe Raymond Garban: But indeed… 664 00:32:01.970 --> 00:32:05.179 Courant Events Right: So, if you are in Z2, you have BKT, Z3, you 665 00:32:05.510 --> 00:32:12.680 Courant Events Right: D4 you don't, and so on, and then H2 strikes back, and then H3 is lost again. So, H2 has a very… 666 00:32:13.070 --> 00:32:13.560 Christophe Raymond Garban: Guys. 667 00:32:13.560 --> 00:32:14.630 Courant Events Right: economic analysis. 668 00:32:14.820 --> 00:32:15.180 Christophe Raymond Garban: Particularly. 669 00:32:15.180 --> 00:32:16.770 Courant Events Right: reality that, there are… 670 00:32:17.080 --> 00:32:17.940 Christophe Raymond Garban: many ways. 671 00:32:17.940 --> 00:32:20.240 Courant Events Right: to virtual harmonic functions. 672 00:32:20.630 --> 00:32:21.300 Christophe Raymond Garban: Real quick. 673 00:32:24.440 --> 00:32:24.990 Christophe Raymond Garban: Go ahead. 674 00:32:24.990 --> 00:32:27.860 Courant Events Right: Beta effective is equal to beta in this case. 675 00:32:28.350 --> 00:32:29.200 Christophe Raymond Garban: Guess. 676 00:32:32.660 --> 00:32:35.410 Courant Events Right: Oh, that's a good question. I should know, I don't… 677 00:32:36.400 --> 00:32:37.850 Christophe Raymond Garban: I don't, I don't… 678 00:32:37.850 --> 00:32:39.359 Courant Events Right: So, but it's a good question. 679 00:32:39.880 --> 00:32:40.450 Courant Events Right: So. 680 00:32:40.450 --> 00:32:41.070 Christophe Raymond Garban: So, the question… 681 00:32:41.070 --> 00:32:46.570 Courant Events Right: by Renault is, is the beta effective the same as the beta here, as opposed to two dimensions? 682 00:32:47.230 --> 00:32:48.969 Christophe Raymond Garban: So. 683 00:32:48.980 --> 00:32:52.940 Courant Events Right: what do we mean by beta effective? So there will be a… a G. 684 00:32:52.940 --> 00:32:53.420 Christophe Raymond Garban: That's true. 685 00:32:53.460 --> 00:32:54.619 Courant Events Right: indexed by. 686 00:32:54.620 --> 00:32:55.150 Christophe Raymond Garban: Yeah, the vertex. 687 00:32:55.150 --> 00:32:59.600 Courant Events Right: of the hyperbolic graph. I still think restricting it to the. 688 00:32:59.600 --> 00:33:02.170 Christophe Raymond Garban: The integers will be lower. 689 00:33:02.630 --> 00:33:03.350 Christophe Raymond Garban: it's… 690 00:33:03.350 --> 00:33:06.049 Courant Events Right: We've been discussing a lot this year together. 691 00:33:06.050 --> 00:33:08.720 Christophe Raymond Garban: and every time you were right, so I don't want to… 692 00:33:09.210 --> 00:33:12.030 Christophe Raymond Garban: Wants to go too far. 693 00:33:12.190 --> 00:33:14.550 Christophe Raymond Garban: So, I want to. 694 00:33:14.550 --> 00:33:18.689 Courant Events Right: go to, the classical Eisenberg now for. 695 00:33:18.690 --> 00:33:20.110 Christophe Raymond Garban: a short time. 696 00:33:20.920 --> 00:33:21.430 Christophe Raymond Garban: And… 697 00:33:21.430 --> 00:33:23.599 Courant Events Right: So now, every spin carries a. 698 00:33:23.600 --> 00:33:24.400 Christophe Raymond Garban: as. 699 00:33:24.400 --> 00:33:24.750 Courant Events Right: too. 700 00:33:24.750 --> 00:33:27.179 Christophe Raymond Garban: I use PIN, and for Yakov. 701 00:33:27.180 --> 00:33:29.749 Courant Events Right: predicted in 75 that. 702 00:33:30.090 --> 00:33:31.130 Christophe Raymond Garban: the… 703 00:33:31.130 --> 00:33:33.120 Courant Events Right: Should have exponential decay. 704 00:33:33.120 --> 00:33:35.349 Christophe Raymond Garban: At arbitrarily low temperature. 705 00:33:35.950 --> 00:33:37.230 Courant Events Right: In other words, you know. 706 00:33:37.230 --> 00:33:38.200 Christophe Raymond Garban: What's clean. 707 00:33:38.200 --> 00:33:42.990 Courant Events Right: Polyakov predicted that this BKT phenomenon that I just explained for the S1. 708 00:33:42.990 --> 00:33:45.290 Christophe Raymond Garban: unvalued spin system does not. 709 00:33:45.290 --> 00:33:48.170 Courant Events Right: when you go to S2-valued spin system. 710 00:33:48.780 --> 00:33:58.869 Courant Events Right: You might say that if you believe this prediction, maybe this spin system is as boring as the one-dimensional easing model, because easing model also has exponential decay. 711 00:33:59.270 --> 00:34:01.399 Christophe Raymond Garban: At all temperature. It's not… 712 00:34:01.400 --> 00:34:02.000 Courant Events Right: really boring. 713 00:34:02.000 --> 00:34:03.430 Christophe Raymond Garban: Because if you solve this. 714 00:34:03.430 --> 00:34:05.649 Courant Events Right: It's considered for many good reasons. 715 00:34:05.950 --> 00:34:06.889 Christophe Raymond Garban: A great story. 716 00:34:06.890 --> 00:34:08.530 Courant Events Right: model to understand confinement. 717 00:34:08.530 --> 00:34:09.050 Christophe Raymond Garban: input. 718 00:34:09.050 --> 00:34:10.670 Courant Events Right: for SU3. 719 00:34:10.670 --> 00:34:11.229 Christophe Raymond Garban: It's still here. 720 00:34:11.230 --> 00:34:12.260 Courant Events Right: Z4, which. 721 00:34:12.260 --> 00:34:15.460 Christophe Raymond Garban: Many people in the audience are more exposed than I am. 722 00:34:16.010 --> 00:34:17.460 Christophe Raymond Garban: So, I want to focus… 723 00:34:17.829 --> 00:34:25.869 Courant Events Right: very shortly on the physics argument here. I will go quickly for two reasons. I'm running a bit off time for what I want to say next. 724 00:34:26.179 --> 00:34:27.760 Christophe Raymond Garban: And also, there are really… 725 00:34:27.760 --> 00:34:28.940 Courant Events Right: bigger experts than. 726 00:34:29.650 --> 00:34:36.270 Christophe Raymond Garban: So, how did Paul Yakov, Understood, that's a… 727 00:34:36.270 --> 00:34:38.959 Courant Events Right: So there should be exponential decay at whole temperatures. 728 00:34:39.600 --> 00:34:46.099 Courant Events Right: So, it goes as follows. So, you sample the… imagine you sample the S2-valued spin system 729 00:34:46.469 --> 00:34:49.469 Courant Events Right: zit graph, that's the original model I started with. 730 00:34:49.949 --> 00:34:51.399 Christophe Raymond Garban: And maybe you want to. 731 00:34:51.400 --> 00:34:57.389 Courant Events Right: grain this a little bit. So maybe you want to represent, that's called the background field method. 732 00:34:57.620 --> 00:34:58.519 Christophe Raymond Garban: You want to represent. 733 00:34:58.520 --> 00:35:06.249 Courant Events Right: represent in which, average direction this box is pointing to, in which average direction this other box is pointing to. 734 00:35:06.650 --> 00:35:07.290 Christophe Raymond Garban: And you… 735 00:35:07.290 --> 00:35:12.060 Courant Events Right: end up with the signal, a coarse-grained signal, which is living at. 736 00:35:12.060 --> 00:35:12.420 Christophe Raymond Garban: That's pretty… 737 00:35:12.420 --> 00:35:13.630 Courant Events Right: low frequencies. 738 00:35:14.230 --> 00:35:16.959 Courant Events Right: The actual spin system in S2. 739 00:35:16.960 --> 00:35:18.039 Christophe Raymond Garban: Who is, in some sense. 740 00:35:18.040 --> 00:35:19.630 Courant Events Right: It's a combination of these. 741 00:35:19.630 --> 00:35:24.499 Christophe Raymond Garban: low-frequency signal, this false grain S2 value, with a higher frequency 742 00:35:24.920 --> 00:35:27.409 Christophe Raymond Garban: which tells you the details of where the. 743 00:35:27.410 --> 00:35:29.709 Courant Events Right: S2 spin system lives near the. 744 00:35:29.710 --> 00:35:33.070 Christophe Raymond Garban: is, slow mode, S2 valued. 745 00:35:33.900 --> 00:35:35.929 Courant Events Right: And now you need this, 746 00:35:35.930 --> 00:35:40.259 Christophe Raymond Garban: business of RGO in this background filters, you try to. 747 00:35:40.260 --> 00:35:44.299 Courant Events Right: average out what the small scales are doing, you try to average out what the. 748 00:35:44.570 --> 00:35:47.299 Christophe Raymond Garban: Higher frequency mode, army. 749 00:35:47.300 --> 00:35:51.309 Courant Events Right: And try to see what you end up with on the, 750 00:35:51.310 --> 00:35:53.100 Christophe Raymond Garban: post-grained L2. 751 00:35:53.100 --> 00:35:53.460 Courant Events Right: consistent. 752 00:35:54.230 --> 00:35:55.050 Christophe Raymond Garban: And what… 753 00:35:55.050 --> 00:35:57.109 Courant Events Right: As you notice, I always try. 754 00:35:57.110 --> 00:36:02.190 Christophe Raymond Garban: But what you eventually notice is that, if you remove lots of. 755 00:36:02.190 --> 00:36:02.980 Courant Events Right: higher order. 756 00:36:02.980 --> 00:36:06.480 Christophe Raymond Garban: After one step of the algae flow. 757 00:36:06.480 --> 00:36:16.650 Courant Events Right: putting aside these higher-order terms, the beta that you have in front of this coarse-grained S2-valued spin system is slightly smaller than the one you thought. 758 00:36:16.650 --> 00:36:17.260 Christophe Raymond Garban: Technically. 759 00:36:17.870 --> 00:36:20.270 Christophe Raymond Garban: And that's how, I think, for Yakos. 760 00:36:20.270 --> 00:36:21.640 Courant Events Right: predicted that if you keep doing. 761 00:36:21.640 --> 00:36:22.729 Christophe Raymond Garban: this RG fluid. 762 00:36:22.730 --> 00:36:24.290 Courant Events Right: To go to higher and higher scale. 763 00:36:24.800 --> 00:36:28.820 Courant Events Right: Eventually, this effective beta will become smaller and smaller. 764 00:36:29.110 --> 00:36:33.730 Courant Events Right: And we'll enter the regime where it's easy to find exponential decay. 765 00:36:34.520 --> 00:36:37.409 Courant Events Right: As I said, I don't have a mass counterpart. 766 00:36:37.410 --> 00:36:39.060 Christophe Raymond Garban: After this prediction. 767 00:36:39.660 --> 00:36:41.030 Christophe Raymond Garban: It's my favorite conjecture. 768 00:36:41.030 --> 00:36:41.500 Courant Events Right: true. 769 00:36:41.500 --> 00:36:42.080 Christophe Raymond Garban: It'. 770 00:36:42.080 --> 00:36:42.620 Courant Events Right: Where's… 771 00:36:42.620 --> 00:36:44.140 Christophe Raymond Garban: conjecturing the field. 772 00:36:44.220 --> 00:36:46.230 Courant Events Right: But I want to go to the seventh part. 773 00:36:46.230 --> 00:36:53.459 Christophe Raymond Garban: where I want to discuss the following question. If you take the polyapos model, what happened? 774 00:36:53.460 --> 00:36:57.670 Courant Events Right: If you slightly nurture the Polyakov portal. So, the Polyakov will be the. 775 00:36:57.670 --> 00:37:00.660 Christophe Raymond Garban: range. I mean, the range will never appear again in the. 776 00:37:00.660 --> 00:37:01.010 Courant Events Right: Right? 777 00:37:01.520 --> 00:37:05.710 Courant Events Right: But the perturbation will either go to the lemon or to the flat pitch. 778 00:37:06.620 --> 00:37:08.560 Christophe Raymond Garban: So… Before we go… 779 00:37:08.560 --> 00:37:11.099 Courant Events Right: the lemon and the flat peach. I want to… 780 00:37:11.380 --> 00:37:21.919 Christophe Raymond Garban: described two, I think, natural perturbations of the classical Eisenberg model. The first natural perturbation, which I think would be the first that. 781 00:37:22.290 --> 00:37:31.159 Courant Events Right: in the room would agree with, would be to say that we could slightly protrude the Romanian metric that we equip the sphere with. 782 00:37:31.600 --> 00:37:33.789 Courant Events Right: So this would be the wrong metric. 783 00:37:34.110 --> 00:37:34.900 Christophe Raymond Garban: I would still… 784 00:37:34.900 --> 00:37:38.360 Courant Events Right: view my spin system as an S2 valued spin system. 785 00:37:38.760 --> 00:37:44.819 Courant Events Right: But the metric I would use to build a Gibbs measure would be the push forward. 786 00:37:44.820 --> 00:37:47.459 Christophe Raymond Garban: from a slightly padded. 787 00:37:47.460 --> 00:37:48.420 Courant Events Right: ape. 788 00:37:49.460 --> 00:37:51.760 Christophe Raymond Garban: So, if you slightly contribute. 789 00:37:51.760 --> 00:37:55.289 Courant Events Right: The metric with epsilon, and then it sends beta to infinity. 790 00:37:55.890 --> 00:37:57.770 Christophe Raymond Garban: We conjecture with. 791 00:37:57.770 --> 00:37:58.899 Courant Events Right: my course. 792 00:37:58.900 --> 00:38:01.310 Christophe Raymond Garban: That's… 793 00:38:01.310 --> 00:38:06.269 Courant Events Right: At least if you have enough symmetry, but we believe maybe this is not even. 794 00:38:06.700 --> 00:38:07.130 Christophe Raymond Garban: This is so… 795 00:38:07.130 --> 00:38:09.730 Courant Events Right: But just to make the conjecture a bit more natural. 796 00:38:09.730 --> 00:38:11.950 Christophe Raymond Garban: We conjecture that it's the. 797 00:38:11.950 --> 00:38:16.579 Courant Events Right: The phenomenology of Polyakov still prevailed, you would still have exponential decay. 798 00:38:16.730 --> 00:38:18.430 Christophe Raymond Garban: At arbitrarily large bids. 799 00:38:18.430 --> 00:38:18.960 Courant Events Right: Yep. 800 00:38:19.630 --> 00:38:21.209 Courant Events Right: Why do we conjecture that? 801 00:38:21.210 --> 00:38:21.790 Christophe Raymond Garban: F? 802 00:38:21.790 --> 00:38:32.400 Courant Events Right: This comes from the fact that there is a work after Polyakov, which is very beautiful. I don't do so much of it even, though I asked Slava to. 803 00:38:32.400 --> 00:38:33.120 Christophe Raymond Garban: Help. 804 00:38:33.120 --> 00:38:42.069 Courant Events Right: me, and this was a huge help, but I still struggle very much. So Dan Friedan, in 1980, he came up with an understanding of the. 805 00:38:43.010 --> 00:38:47.599 Courant Events Right: renormalization group flow of more general spin systems than classical Eisenberg. 806 00:38:47.600 --> 00:38:48.200 Christophe Raymond Garban: Okay? 807 00:38:48.220 --> 00:38:50.450 Courant Events Right: Precisely of those spin systems here. 808 00:38:50.970 --> 00:38:54.179 Christophe Raymond Garban: And he found out that along the RG flow. 809 00:38:54.180 --> 00:38:58.490 Courant Events Right: The relevant thing to follow is the rich sea flow of this, 810 00:38:58.490 --> 00:39:00.260 Christophe Raymond Garban: Romanian metric. 811 00:39:00.400 --> 00:39:05.999 Courant Events Right: And if I slightly perturb the round metric, it will flow to the wrong sphere quicker. 812 00:39:06.000 --> 00:39:06.390 Christophe Raymond Garban: than. 813 00:39:06.390 --> 00:39:08.369 Courant Events Right: and the way beta is evolving. 814 00:39:08.440 --> 00:39:15.120 Christophe Raymond Garban: So, if one takes that, I think Slava agreed, but, but… 815 00:39:15.120 --> 00:39:21.199 Courant Events Right: as, Schenko, disagreed, so I cannot blame either of the two. It was short discussions. 816 00:39:21.400 --> 00:39:23.260 Courant Events Right: But we conjecture. 817 00:39:23.260 --> 00:39:25.520 Christophe Raymond Garban: the Richie Flow gives the right. 818 00:39:25.780 --> 00:39:31.280 Courant Events Right: Beautiful, and that's… The Polyakov phenomenology is exactly the same for this perturbation. 819 00:39:31.580 --> 00:39:37.440 Courant Events Right: The second perturbation I will now, explain to you is more this lemon and the flat peach. 820 00:39:37.770 --> 00:39:39.470 Courant Events Right: I should have changed my slide. 821 00:39:39.470 --> 00:39:42.099 Christophe Raymond Garban: So, another natural. 822 00:39:42.100 --> 00:39:45.799 Courant Events Right: observation that you could come up with, is instead of changing the metric. 823 00:39:45.800 --> 00:39:46.410 Christophe Raymond Garban: Thank you. 824 00:39:46.410 --> 00:39:51.530 Courant Events Right: Maybe you want to change the Scala product that I used before to define the Gibbs measure. 825 00:39:52.270 --> 00:39:53.370 Christophe Raymond Garban: So here, the… 826 00:39:53.370 --> 00:39:54.060 Courant Events Right: The Sigmas. 827 00:39:54.060 --> 00:39:54.650 Christophe Raymond Garban: attached. 828 00:39:54.650 --> 00:39:57.949 Courant Events Right: to each vertex of Z2. I will write those in. 829 00:39:57.950 --> 00:40:00.099 Christophe Raymond Garban: in the coordinates of R3. 830 00:40:00.100 --> 00:40:02.030 Courant Events Right: Sigma 1, Sigma 2, Sigma 3. 831 00:40:02.620 --> 00:40:03.560 Christophe Raymond Garban: If London… 832 00:40:03.560 --> 00:40:09.239 Courant Events Right: is equal to 1, this is just exactly the classical Eisenberg model I played with up to now. 833 00:40:09.560 --> 00:40:10.880 Christophe Raymond Garban: But I want to slightly. 834 00:40:10.880 --> 00:40:13.110 Courant Events Right: change the strength of the third company. 835 00:40:13.110 --> 00:40:14.300 Christophe Raymond Garban: Which… 836 00:40:14.300 --> 00:40:16.399 Courant Events Right: In my pictures will be the vertical component. 837 00:40:17.280 --> 00:40:20.710 Christophe Raymond Garban: So if lambda is larger, I'm basically, 838 00:40:20.710 --> 00:40:23.940 Courant Events Right: a Lemon-valued spin system, which puts more. 839 00:40:23.940 --> 00:40:24.420 Christophe Raymond Garban: Cool! 840 00:40:24.420 --> 00:40:28.560 Courant Events Right: strength to spins pointing in the north or south. 841 00:40:28.560 --> 00:40:29.340 Christophe Raymond Garban: direction. 842 00:40:29.470 --> 00:40:30.610 Christophe Raymond Garban: Longdae is equal. 843 00:40:30.610 --> 00:40:33.060 Courant Events Right: One is the perfect orange, and this is the. 844 00:40:33.060 --> 00:40:34.400 Christophe Raymond Garban: the flat speech. 845 00:40:35.740 --> 00:40:38.850 Courant Events Right: So, what are the known results here? If lambda is. 846 00:40:38.850 --> 00:40:39.890 Christophe Raymond Garban: is, 847 00:40:39.890 --> 00:40:40.480 Courant Events Right: large. 848 00:40:40.750 --> 00:40:43.549 Courant Events Right: Boats and griffids have shown in the 70s. 849 00:40:43.870 --> 00:40:46.950 Christophe Raymond Garban: That at low enough temperature, the spin system will carry. 850 00:40:46.950 --> 00:40:47.760 Courant Events Right: such an easing phase. 851 00:40:47.760 --> 00:40:48.719 Christophe Raymond Garban: It didn't true. 852 00:40:48.860 --> 00:40:49.410 Courant Events Right: Be either. 853 00:40:49.410 --> 00:40:52.919 Christophe Raymond Garban: very close to the North Pole, or very close to the South Pole. 854 00:40:53.600 --> 00:40:54.789 Christophe Raymond Garban: The gap was… 855 00:40:54.790 --> 00:41:02.290 Courant Events Right: filled by Mali Sheff and Froli Schlieb in the late 70s, where they proved that this is still true. 856 00:41:02.580 --> 00:41:03.250 Christophe Raymond Garban: For one letter. 857 00:41:03.250 --> 00:41:05.079 Courant Events Right: I mean, arbitrary growth to the. 858 00:41:05.080 --> 00:41:05.750 Christophe Raymond Garban: Jewish. 859 00:41:06.690 --> 00:41:08.530 Christophe Raymond Garban: And in this work by Felicia. 860 00:41:08.530 --> 00:41:10.780 Courant Events Right: They introduced this, 861 00:41:10.780 --> 00:41:13.200 Christophe Raymond Garban: chessboard estimate, which happened. 862 00:41:13.200 --> 00:41:15.110 Courant Events Right: Could be very useful since then. 863 00:41:15.960 --> 00:41:23.899 Courant Events Right: So now, I will cross over to the flat pitch, and brickman Levovitzpister, they conjecture that if you go to the flat pitch. 864 00:41:24.210 --> 00:41:25.999 Courant Events Right: At any longer less than what? 865 00:41:26.000 --> 00:41:27.669 Christophe Raymond Garban: If you slightly perturb the. 866 00:41:27.670 --> 00:41:28.319 Courant Events Right: cool as in the. 867 00:41:28.320 --> 00:41:32.439 Christophe Raymond Garban: Now, a BKT phase should, should erupt. 868 00:41:33.360 --> 00:41:37.979 Courant Events Right: In fact, while they conjectured this, they actually proved the conjecture was wrong. 869 00:41:37.980 --> 00:41:42.169 Christophe Raymond Garban: 1 octa is exactly equal to zero, which doesn't give any. 870 00:41:42.170 --> 00:41:43.000 Courant Events Right: strengths to the. 871 00:41:43.000 --> 00:41:44.190 Christophe Raymond Garban: critical direction. 872 00:41:44.840 --> 00:41:47.790 Courant Events Right: Soon after, Dunlop proved that if. 873 00:41:47.790 --> 00:41:52.540 Christophe Raymond Garban: The longer is small enough, if you take a very flat pitch, if you just push hard on your pitch. 874 00:41:53.070 --> 00:42:00.410 Courant Events Right: Then, indeed, at very low temperature, the spin system will experience a power loop decay phase along the equator here. 875 00:42:01.570 --> 00:42:02.090 Christophe Raymond Garban: So… 876 00:42:02.090 --> 00:42:07.320 Courant Events Right: Now comes our main CRM with Nathan de Montgolfie. What we can show is that. 877 00:42:07.600 --> 00:42:12.729 Christophe Raymond Garban: For arbitrarily small perturbation of the classical eigen parameter in this subconscious. 878 00:42:12.730 --> 00:42:13.080 Courant Events Right: sense. 879 00:42:13.560 --> 00:42:14.530 Christophe Raymond Garban: Because in the first. 880 00:42:14.530 --> 00:42:17.220 Courant Events Right: In other sense, we conjecture it's a different phenomenology. 881 00:42:17.320 --> 00:42:19.110 Courant Events Right: Then, yes, a decade phase. 882 00:42:19.110 --> 00:42:19.470 Christophe Raymond Garban: is… 883 00:42:19.470 --> 00:42:20.250 Courant Events Right: 16. 884 00:42:21.550 --> 00:42:23.880 Courant Events Right: Here is the second theorem, which has some. 885 00:42:23.880 --> 00:42:24.490 Christophe Raymond Garban: some, 886 00:42:24.490 --> 00:42:25.319 Courant Events Right: quotation marks? 887 00:42:25.320 --> 00:42:26.369 Christophe Raymond Garban: I will explain what's. 888 00:42:26.370 --> 00:42:27.050 Courant Events Right: going on here. 889 00:42:27.590 --> 00:42:28.460 Christophe Raymond Garban: So… 890 00:42:28.460 --> 00:42:30.890 Courant Events Right: Since we have many physicists in the room. 891 00:42:31.160 --> 00:42:31.630 Christophe Raymond Garban: It… 892 00:42:31.630 --> 00:42:36.649 Courant Events Right: It may be interesting to ask what the RG flow is doing on this perturbation. 893 00:42:36.930 --> 00:42:44.519 Courant Events Right: So maybe it's natural to try to plot in the plane where I have the temperature here, and I have the amount of perturbation in the. 894 00:42:44.850 --> 00:42:46.060 Christophe Raymond Garban: vertical. 895 00:42:46.060 --> 00:42:46.420 Courant Events Right: excuse? 896 00:42:46.790 --> 00:42:47.810 Christophe Raymond Garban: What's going on? 897 00:42:47.810 --> 00:42:55.920 Courant Events Right: What is the phase which would be exponentially decaying? What is the phase which would be long-range order and easing light? And what is the phase which would be deciding? 898 00:42:56.650 --> 00:42:57.970 Christophe Raymond Garban: Of course, we do not. 899 00:42:57.970 --> 00:43:00.130 Courant Events Right: Prove where the lines are exactly. 900 00:43:00.130 --> 00:43:03.710 Christophe Raymond Garban: But if we were to prove this, it would be more difficult than polyaco. 901 00:43:03.710 --> 00:43:04.590 Courant Events Right: conjunctional. 902 00:43:04.590 --> 00:43:05.089 Christophe Raymond Garban: for you, because. 903 00:43:05.090 --> 00:43:05.510 Courant Events Right: predictions. 904 00:43:06.120 --> 00:43:07.460 Christophe Raymond Garban: But the first part of. 905 00:43:07.460 --> 00:43:08.270 Courant Events Right: these. 906 00:43:08.270 --> 00:43:11.829 Christophe Raymond Garban: to M is actually not our contribution, it's, 907 00:43:11.830 --> 00:43:18.910 Courant Events Right: Slava during a lunch at IHCS on pieces of paper this big, where he helped us understanding what. 908 00:43:18.910 --> 00:43:21.629 Christophe Raymond Garban: could be the curve here, along the RG flow. 909 00:43:22.210 --> 00:43:26.959 Courant Events Right: So, we'll try to explain this in the paper, as best as we can. 910 00:43:27.290 --> 00:43:28.140 Christophe Raymond Garban: But where we. 911 00:43:28.140 --> 00:43:30.670 Courant Events Right: We do have a CRM, he's in the… so that's… 912 00:43:30.670 --> 00:43:33.739 Christophe Raymond Garban: supposedly the RG flow, What we are doing. 913 00:43:33.740 --> 00:43:34.899 Courant Events Right: I mean, the second part. 914 00:43:34.900 --> 00:43:35.940 Christophe Raymond Garban: of this paper. 915 00:43:35.940 --> 00:43:37.839 Courant Events Right: We take, 916 00:43:37.840 --> 00:43:39.119 Christophe Raymond Garban: like a space where exposition. 917 00:43:39.120 --> 00:43:39.920 Courant Events Right: financial decay. 918 00:43:40.050 --> 00:43:40.820 Christophe Raymond Garban: is actually… 919 00:43:40.820 --> 00:43:42.690 Courant Events Right: be known rigorously. 920 00:43:42.940 --> 00:43:44.289 Christophe Raymond Garban: And this is in a work by. 921 00:43:44.290 --> 00:43:46.260 Courant Events Right: Boris Schmidt, Enmuss, and Schwab. 922 00:43:46.260 --> 00:43:47.720 Christophe Raymond Garban: Where they. 923 00:43:47.720 --> 00:43:53.340 Courant Events Right: show that if you take the hemisphere, so there's no SO3 symmetry, but still you take the hemisphere. 924 00:43:53.670 --> 00:43:54.959 Christophe Raymond Garban: There is exponential. 925 00:43:54.960 --> 00:43:55.630 Courant Events Right: ridiculous. 926 00:43:55.630 --> 00:43:57.360 Christophe Raymond Garban: population. In the two core. 927 00:43:57.360 --> 00:44:00.149 Courant Events Right: Coordinates that are along the plane here. 928 00:44:00.880 --> 00:44:03.299 Courant Events Right: And so this is known, this is their CRM. 929 00:44:03.560 --> 00:44:04.330 Christophe Raymond Garban: And what we. 930 00:44:04.330 --> 00:44:04.989 Courant Events Right: do with Netflix. 931 00:44:04.990 --> 00:44:10.349 Christophe Raymond Garban: We try to understand up to which deformation of the atmosphere we can push. 932 00:44:10.350 --> 00:44:14.999 Courant Events Right: the exponential decay phase, and the good news We can reach the, 933 00:44:15.000 --> 00:44:17.290 Christophe Raymond Garban: the line that's lava and Earth. 934 00:44:17.290 --> 00:44:19.140 Courant Events Right: to understand with the RG flow. 935 00:44:19.790 --> 00:44:20.660 Courant Events Right: So, at least… 936 00:44:20.660 --> 00:44:21.240 Christophe Raymond Garban: At least for the. 937 00:44:21.240 --> 00:44:22.909 Courant Events Right: Hemisphere, we do have a regular. 938 00:44:22.910 --> 00:44:25.120 Christophe Raymond Garban: explanation of these are Gila. 939 00:44:26.120 --> 00:44:27.460 Christophe Raymond Garban: So, in the essence of… 940 00:44:27.460 --> 00:44:30.300 Courant Events Right: In this talk, what I want to do is to explain to. 941 00:44:30.300 --> 00:44:32.659 Christophe Raymond Garban: you, how do we prove. 942 00:44:32.660 --> 00:44:33.440 Courant Events Right: CRM1. 943 00:44:33.800 --> 00:44:36.180 Courant Events Right: And I will do it in two steps. 944 00:44:36.430 --> 00:44:38.700 Courant Events Right: The first step, I will tell you how to. 945 00:44:38.700 --> 00:44:39.450 Christophe Raymond Garban: approve? 946 00:44:39.450 --> 00:44:43.390 Courant Events Right: Why a very flattened beach must have a big ediface. 947 00:44:43.890 --> 00:44:45.789 Courant Events Right: the temperature is small, so I. 948 00:44:45.790 --> 00:44:48.310 Christophe Raymond Garban: improve the Lobe CRM. 949 00:44:48.650 --> 00:44:49.410 Christophe Raymond Garban: And then… 950 00:44:49.410 --> 00:44:54.360 Courant Events Right: I will want to explain to you why there is a barrier, why there is a long dazzero. 951 00:44:54.360 --> 00:45:00.269 Christophe Raymond Garban: of which we need to do something else. And the second step will require properties of a 952 00:45:00.670 --> 00:45:02.240 Christophe Raymond Garban: I want to. 953 00:45:02.240 --> 00:45:04.730 Courant Events Right: maps that are a well-studied object in. 954 00:45:04.730 --> 00:45:05.730 Christophe Raymond Garban: analyses. 955 00:45:05.730 --> 00:45:09.569 Courant Events Right: And why do we need harmonic maps? To close the gap and go all the way. 956 00:45:09.570 --> 00:45:11.349 Christophe Raymond Garban: Way to the orange. 957 00:45:12.760 --> 00:45:15.450 Christophe Raymond Garban: So, the first thing I do here is I recall what. 958 00:45:15.450 --> 00:45:16.279 Courant Events Right: the motor here? 959 00:45:16.560 --> 00:45:17.960 Christophe Raymond Garban: My spins are. 960 00:45:17.960 --> 00:45:28.599 Courant Events Right: two valued, I write those in those three coordinates of R3, and lambda tells you how much strength do I give to the third coordinate. So the Gibbs measure. 961 00:45:28.600 --> 00:45:32.530 Christophe Raymond Garban: is proportional to exponential beta sums over this. 962 00:45:32.530 --> 00:45:36.059 Courant Events Right: modified scalar products between neighboring sites. 963 00:45:37.180 --> 00:45:37.700 Courant Events Right: Different? 964 00:45:37.700 --> 00:45:38.189 Christophe Raymond Garban: First, I… 965 00:45:38.190 --> 00:45:41.570 Courant Events Right: ID that we use is the following. It's a classical ID, which 966 00:45:41.820 --> 00:45:43.540 Courant Events Right: Bows back, maybe at least to. 967 00:45:43.590 --> 00:45:44.250 Christophe Raymond Garban: But how… 968 00:45:44.250 --> 00:45:45.680 Courant Events Right: for you, Xyler, I will say a word. 969 00:45:45.680 --> 00:45:46.700 Christophe Raymond Garban: about that. 970 00:45:46.920 --> 00:45:49.549 Courant Events Right: So, the idea that they had is that they can. 971 00:45:49.830 --> 00:45:50.950 Christophe Raymond Garban: Look at this! 972 00:45:50.950 --> 00:45:51.689 Courant Events Right: two valued. 973 00:45:51.690 --> 00:45:53.780 Christophe Raymond Garban: system, and they can ask about the. 974 00:45:53.780 --> 00:45:55.130 Courant Events Right: brewing, natural. 975 00:45:55.130 --> 00:45:59.679 Christophe Raymond Garban: polaristic question. Conditioned on the vertical direction, on the. 976 00:45:59.680 --> 00:46:00.900 Courant Events Right: Sigma 3 field. 977 00:46:01.200 --> 00:46:03.609 Christophe Raymond Garban: What is the law of the remaining randomness? 978 00:46:03.840 --> 00:46:06.049 Courant Events Right: So, probabilistic-wise, the sample. 979 00:46:06.050 --> 00:46:07.310 Christophe Raymond Garban: the third. 980 00:46:07.310 --> 00:46:12.240 Courant Events Right: coordinate. This Pilometer measure is the measure we are interested in. It's a complicated measure. 981 00:46:13.110 --> 00:46:18.790 Courant Events Right: And I want to understand what is the remaining randomness if you give to me the positions. 982 00:46:19.060 --> 00:46:22.949 Christophe Raymond Garban: the vertical positions on the S2 field. 983 00:46:23.360 --> 00:46:27.349 Christophe Raymond Garban: And the observation, which goes back to Patrasios. 984 00:46:27.350 --> 00:46:28.609 Courant Events Right: Even older, I don't know. 985 00:46:28.920 --> 00:46:31.120 Christophe Raymond Garban: Is that the condition of the. 986 00:46:31.120 --> 00:46:32.820 Courant Events Right: Remaining randomness to sample. 987 00:46:32.920 --> 00:46:34.420 Christophe Raymond Garban: It's extremely simple. 988 00:46:34.420 --> 00:46:34.880 Courant Events Right: to run. 989 00:46:34.880 --> 00:46:41.560 Christophe Raymond Garban: It's exactly an XY model, as we had in the first part of this talk. The only difference is that it. 990 00:46:41.560 --> 00:46:45.659 Courant Events Right: It's an XY model in a random disorder of conductances. 991 00:46:45.940 --> 00:46:48.120 Courant Events Right: The conductance from this point to this point. 992 00:46:48.120 --> 00:46:49.330 Christophe Raymond Garban: is beta? 993 00:46:49.330 --> 00:46:54.580 Courant Events Right: times the radius here for the… for the S1 spin, times the radius here. 994 00:46:54.710 --> 00:46:56.800 Courant Events Right: So it's very simple to describe. 995 00:46:57.120 --> 00:46:59.450 Christophe Raymond Garban: condition on this, and using pedagogy. 996 00:46:59.450 --> 00:47:00.040 Courant Events Right: SUM and. 997 00:47:00.040 --> 00:47:00.500 Christophe Raymond Garban: Nothing ahead. 998 00:47:00.500 --> 00:47:02.010 Courant Events Right: What is the fee? 999 00:47:02.010 --> 00:47:03.330 Christophe Raymond Garban: In the random conductance. 1000 00:47:03.330 --> 00:47:03.940 Courant Events Right: where… 1001 00:47:04.150 --> 00:47:07.200 Christophe Raymond Garban: the, the spin in S1 is. 1002 00:47:07.200 --> 00:47:08.370 Courant Events Right: It still needs to be sampled. 1003 00:47:09.100 --> 00:47:12.319 Christophe Raymond Garban: So, if you do Polyakov conjecture, that was what. 1004 00:47:12.320 --> 00:47:15.219 Courant Events Right: used to try to argue against. 1005 00:47:15.220 --> 00:47:15.700 Christophe Raymond Garban: Yeah, good. 1006 00:47:16.470 --> 00:47:17.960 Courant Events Right: A few years ago. 1007 00:47:17.960 --> 00:47:18.330 Christophe Raymond Garban: Who is. 1008 00:47:18.330 --> 00:47:20.129 Courant Events Right: You are here and have a UC pouveau. 1009 00:47:20.130 --> 00:47:25.380 Christophe Raymond Garban: we re-discussed Patrice Allure, and we basically gave a counterexample. 1010 00:47:25.380 --> 00:47:29.940 Courant Events Right: to their argument against Polyakov. Of course, we did not prove Polyakov's prediction at all. 1011 00:47:30.250 --> 00:47:30.819 Christophe Raymond Garban: But we. 1012 00:47:30.820 --> 00:47:34.490 Courant Events Right: provided a counterexample to the analysis in the 90s, which I think. 1013 00:47:35.080 --> 00:47:35.830 Christophe Raymond Garban: businesses… 1014 00:47:35.830 --> 00:47:36.290 Courant Events Right: didn't… 1015 00:47:36.290 --> 00:47:38.040 Christophe Raymond Garban: Bave the analysis anyway. 1016 00:47:38.630 --> 00:47:42.219 Courant Events Right: I really liked the idea to try to test how robust. 1017 00:47:42.220 --> 00:47:44.220 Christophe Raymond Garban: parallel decay could be. 1018 00:47:44.220 --> 00:47:47.170 Courant Events Right: in this, projected. 1019 00:47:47.170 --> 00:47:47.729 Christophe Raymond Garban: Thank you. 1020 00:47:48.090 --> 00:47:53.920 Christophe Raymond Garban: In this work, we definitely go through Patraska's ILRID, but this time we get a series. 1021 00:47:54.340 --> 00:47:59.060 Christophe Raymond Garban: Because we get some control on this random disorder when we project. 1022 00:47:59.610 --> 00:48:04.420 Courant Events Right: the, flat pitch system down to an XY. 1023 00:48:05.200 --> 00:48:07.090 Courant Events Right: So, in. 1024 00:48:07.890 --> 00:48:08.520 Christophe Raymond Garban: But find me. 1025 00:48:08.520 --> 00:48:09.080 Courant Events Right: it? 1026 00:48:12.930 --> 00:48:15.620 Courant Events Right: I think I do. 1027 00:48:16.810 --> 00:48:18.359 Courant Events Right: We started at 20. 1028 00:48:19.350 --> 00:48:20.370 Courant Events Right: It's sense. 1029 00:48:20.630 --> 00:48:23.310 Christophe Raymond Garban: So, according to… what? 1030 00:48:23.310 --> 00:48:23.740 Courant Events Right: we've been. 1031 00:48:23.740 --> 00:48:28.049 Christophe Raymond Garban: Right, so… No, no, no, no, I… 1032 00:48:28.050 --> 00:48:29.890 Courant Events Right: That's right. Yes. 1033 00:48:29.900 --> 00:48:30.710 Christophe Raymond Garban: Last question. 1034 00:48:31.190 --> 00:48:32.460 Courant Events Right: I, I think 5. 1035 00:48:32.460 --> 00:48:36.890 Christophe Raymond Garban: I haven't… So, the idea. 1036 00:48:36.890 --> 00:48:42.629 Courant Events Right: Is that this field of conductances is too hard to describe. 1037 00:48:43.260 --> 00:48:43.810 Christophe Raymond Garban: But… 1038 00:48:43.810 --> 00:48:44.630 Courant Events Right: The… the… 1039 00:48:44.990 --> 00:48:51.299 Christophe Raymond Garban: The proof is trying to control this disorder by comparing with a percolation disorder. 1040 00:48:51.720 --> 00:48:58.600 Courant Events Right: So, the idea is to try to say that maybe there is a stochastic domination between this field of conductances. 1041 00:48:58.600 --> 00:49:00.150 Christophe Raymond Garban: And a nicer… 1042 00:49:00.150 --> 00:49:03.530 Courant Events Right: field of conductances, which would be percolation disorder. 1043 00:49:04.300 --> 00:49:05.630 Christophe Raymond Garban: And here, we spoke. 1044 00:49:05.630 --> 00:49:07.339 Courant Events Right: Where are you, a few years? 1045 00:49:07.340 --> 00:49:08.260 Christophe Raymond Garban: years ago. 1046 00:49:08.260 --> 00:49:10.960 Courant Events Right: We prove that if you try to do XY. 1047 00:49:10.960 --> 00:49:12.769 Christophe Raymond Garban: model on a percolation. 1048 00:49:12.770 --> 00:49:13.560 Courant Events Right: disorder. 1049 00:49:14.080 --> 00:49:20.350 Christophe Raymond Garban: Then, at low enough temperature, you will still have quasi-long-range order on such percolating. 1050 00:49:20.350 --> 00:49:20.970 Courant Events Right: the cluster. 1051 00:49:20.970 --> 00:49:23.819 Christophe Raymond Garban: All the way to the population transition. 1052 00:49:24.870 --> 00:49:25.400 Christophe Raymond Garban: 2? 1053 00:49:25.400 --> 00:49:26.620 Courant Events Right: The idea with not only. 1054 00:49:26.620 --> 00:49:28.829 Christophe Raymond Garban: the Mughal fee is that if Longda is. 1055 00:49:28.830 --> 00:49:29.340 Courant Events Right: Very small. 1056 00:49:29.340 --> 00:49:29.950 Christophe Raymond Garban: moon? 1057 00:49:30.230 --> 00:49:31.210 Christophe Raymond Garban: We observed that… 1058 00:49:31.210 --> 00:49:36.110 Courant Events Right: And it's not too difficult to prove that we can stochastically dominate our field of. 1059 00:49:36.110 --> 00:49:37.210 Christophe Raymond Garban: conductances. 1060 00:49:37.210 --> 00:49:40.700 Courant Events Right: with a Bernoulli, with an IID field of disorders. 1061 00:49:41.140 --> 00:49:42.660 Courant Events Right: And the idea is simple. 1062 00:49:42.660 --> 00:49:43.010 Christophe Raymond Garban: I don't. 1063 00:49:43.010 --> 00:49:44.950 Courant Events Right: He wants to have stochastic domination. 1064 00:49:44.950 --> 00:49:55.499 Christophe Raymond Garban: Then you can try to condition on all the spin field outside of one point, and try to prove that whatever you see outside, you still have a high probability. 1065 00:49:55.500 --> 00:49:56.830 Courant Events Right: to be far from North Pole. 1066 00:49:56.830 --> 00:49:57.410 Christophe Raymond Garban: recalls how. 1067 00:49:57.410 --> 00:49:58.010 Courant Events Right: pool. 1068 00:49:58.570 --> 00:50:03.359 Courant Events Right: So, this doesn't work, because if you're unlucky, and the four points around you. 1069 00:50:03.360 --> 00:50:05.549 Christophe Raymond Garban: are close to the North Pole. 1070 00:50:05.550 --> 00:50:08.610 Courant Events Right: Even if you have a very flat pitch, you will still be… 1071 00:50:08.610 --> 00:50:09.889 Christophe Raymond Garban: We want to be very close. 1072 00:50:09.890 --> 00:50:10.549 Courant Events Right: to the North Pole. 1073 00:50:10.550 --> 00:50:11.859 Christophe Raymond Garban: at low temperature. 1074 00:50:12.290 --> 00:50:17.399 Christophe Raymond Garban: So this does not work, but if we make the… if we make it… 1075 00:50:17.400 --> 00:50:21.909 Courant Events Right: slightly bigger. If we now look at the neighborhood of just one edge. 1076 00:50:22.170 --> 00:50:25.439 Courant Events Right: Now we can have… we have a lemma which says that if the pinch. 1077 00:50:25.440 --> 00:50:26.689 Christophe Raymond Garban: It's flat enough. 1078 00:50:26.980 --> 00:50:32.499 Christophe Raymond Garban: Yes, with high probability, uniformly on what we see away from us. 1079 00:50:32.720 --> 00:50:38.450 Christophe Raymond Garban: we will have two spins which will both be far from the North Pole. 1080 00:50:39.540 --> 00:50:41.630 Courant Events Right: to tell you why, imagine you're very. 1081 00:50:41.630 --> 00:50:43.599 Christophe Raymond Garban: and a 6S2, but. 1082 00:50:43.600 --> 00:50:44.720 Courant Events Right: It spins around you are. 1083 00:50:44.720 --> 00:50:46.510 Christophe Raymond Garban: All north pointing spins. 1084 00:50:46.870 --> 00:50:50.659 Christophe Raymond Garban: Because of the flat speech, the system will still want to simulate. 1085 00:50:50.660 --> 00:50:51.280 Courant Events Right: when you sleep on. 1086 00:50:51.280 --> 00:50:58.480 Christophe Raymond Garban: to a direction, and be far to the North Pole. So our argument is based just on the behavior of the. 1087 00:50:58.480 --> 00:51:03.420 Courant Events Right: the harmonic maps, the ground states, uniformly in the neighborhood. 1088 00:51:04.360 --> 00:51:04.970 Courant Events Right: These are… 1089 00:51:04.970 --> 00:51:07.509 Christophe Raymond Garban: The argument is rather soft, but it doesn't work when. 1090 00:51:07.510 --> 00:51:10.499 Courant Events Right: And lambda is bigger than 1 over 100. 1091 00:51:11.280 --> 00:51:13.380 Christophe Raymond Garban: As lambda goes to 1. 1092 00:51:13.380 --> 00:51:16.629 Courant Events Right: It's easy to see that we are forced to analyze. 1093 00:51:16.630 --> 00:51:20.859 Christophe Raymond Garban: situations which go at farther and further radii. 1094 00:51:21.380 --> 00:51:22.000 Christophe Raymond Garban: So, as. 1095 00:51:22.000 --> 00:51:27.050 Courant Events Right: As long back goes to infinity, we need to understand uniformly in what we may see outside of. 1096 00:51:27.050 --> 00:51:30.340 Christophe Raymond Garban: the pool of radius are? How does the harmonic. 1097 00:51:30.340 --> 00:51:35.709 Courant Events Right: maps how does the least energetic pitch-valued function may look. 1098 00:51:36.450 --> 00:51:40.539 Courant Events Right: And being in New York was actually quite useful this year, because. 1099 00:51:40.840 --> 00:51:42.340 Christophe Raymond Garban: I include this. 1100 00:51:42.340 --> 00:51:43.280 Courant Events Right: with offenders. 1101 00:51:43.280 --> 00:51:44.939 Christophe Raymond Garban: and with Naton, we try. 1102 00:51:44.940 --> 00:51:45.850 Courant Events Right: to use. 1103 00:51:46.270 --> 00:51:48.889 Christophe Raymond Garban: Proof techniques that were used in. 1104 00:51:48.890 --> 00:51:55.530 Courant Events Right: Amanda analysis of harmonic maps, and we could show that they're not as ordered, as regular. 1105 00:51:55.530 --> 00:51:59.559 Christophe Raymond Garban: that we were hoping for. If we had regular harmonic functions. 1106 00:51:59.560 --> 00:52:01.250 Courant Events Right: linear program, everything would be very. 1107 00:52:01.250 --> 00:52:02.120 Christophe Raymond Garban: the group. 1108 00:52:02.250 --> 00:52:02.800 Christophe Raymond Garban: You can… 1109 00:52:02.800 --> 00:52:06.089 Courant Events Right: It's less regular, but we have enough weak regularity to conclude. 1110 00:52:06.790 --> 00:52:07.660 Christophe Raymond Garban: And the week… 1111 00:52:07.660 --> 00:52:10.310 Courant Events Right: The configurality that we used is something Sylvia. 1112 00:52:10.310 --> 00:52:13.480 Christophe Raymond Garban: knows very well, it's a Ginsburg-Lando type. 1113 00:52:13.650 --> 00:52:17.440 Christophe Raymond Garban: And plugging this with, the world by. 1114 00:52:17.440 --> 00:52:19.379 Courant Events Right: for Dario to propagate BKT. 1115 00:52:19.670 --> 00:52:20.400 Christophe Raymond Garban: On the bed… 1116 00:52:20.400 --> 00:52:23.189 Courant Events Right: operation disorder, we could handle this arbitrary. 1117 00:52:23.190 --> 00:52:26.820 Christophe Raymond Garban: small percolation of S2, I will solve it. 1118 00:52:26.820 --> 00:52:27.970 Courant Events Right: And… 1119 00:52:27.970 --> 00:52:29.990 Christophe Raymond Garban: Thank you. 1120 00:52:36.150 --> 00:52:39.489 Courant Events Right: Alright, also, we had, already a lately of directions. 1121 00:52:39.490 --> 00:52:41.289 Christophe Raymond Garban: And we can continue with the introduction. 1122 00:52:42.990 --> 00:52:44.120 Christophe Raymond Garban: Authorities. 1123 00:52:47.800 --> 00:52:48.669 Courant Events Right: Do you think. 1124 00:52:48.670 --> 00:52:49.899 Christophe Raymond Garban: productive, so… 1125 00:52:49.900 --> 00:52:51.470 Courant Events Right: clients, or also to teach her. 1126 00:52:51.470 --> 00:52:53.420 Christophe Raymond Garban: We opened every space for the moment. 1127 00:52:53.420 --> 00:52:54.290 Courant Events Right: Where do you think those funds. 1128 00:52:54.290 --> 00:52:55.400 Christophe Raymond Garban: We don't run up. 1129 00:52:55.400 --> 00:52:57.180 Courant Events Right: I've been upon it that they furnished. 1130 00:52:58.890 --> 00:53:01.430 Courant Events Right: yes, it's a good question. 1131 00:53:05.010 --> 00:53:07.890 Courant Events Right: I… I… It's a good quote, I believe so. 1132 00:53:08.540 --> 00:53:10.560 Courant Events Right: So, indeed, if the, 1133 00:53:10.560 --> 00:53:10.910 Christophe Raymond Garban: gear. 1134 00:53:10.910 --> 00:53:12.330 Courant Events Right: Each measure is now. 1135 00:53:12.830 --> 00:53:13.440 Christophe Raymond Garban: Excellent. 1136 00:53:13.440 --> 00:53:16.989 Courant Events Right: financial beta sum of Scala product squares. 1137 00:53:17.150 --> 00:53:19.080 Christophe Raymond Garban: The model is a model of. 1138 00:53:19.080 --> 00:53:21.540 Courant Events Right: Random lines instead of random points on the sphere. 1139 00:53:21.910 --> 00:53:22.899 Christophe Raymond Garban: This is also a. 1140 00:53:22.900 --> 00:53:25.420 Courant Events Right: I believe to have an exponential decay. 1141 00:53:25.420 --> 00:53:26.470 Christophe Raymond Garban: habitual. 1142 00:53:27.140 --> 00:53:28.860 Christophe Raymond Garban: Oh, no, there's busy. 1143 00:53:29.430 --> 00:53:31.280 Courant Events Right: Is this point? 1144 00:53:31.960 --> 00:53:32.890 Christophe Raymond Garban: Secretary. 1145 00:53:33.760 --> 00:53:40.399 Christophe Raymond Garban: Oh, these are very interesting. If this is proof to… if it's believed to be wrong, I won'. 1146 00:53:40.400 --> 00:53:42.559 Courant Events Right: To go on this model and see what's going on. 1147 00:53:47.370 --> 00:53:48.350 Courant Events Right: Can you explain? 1148 00:53:51.570 --> 00:53:53.129 Christophe Raymond Garban: And can you explain this? 1149 00:53:53.130 --> 00:53:56.669 Courant Events Right: How many different variation in the language of the law. 1150 00:53:57.270 --> 00:53:58.149 Christophe Raymond Garban: What is it looking like? 1151 00:53:58.150 --> 00:54:01.410 Courant Events Right: Okay, so. 1152 00:54:01.660 --> 00:54:02.889 Christophe Raymond Garban: So the… 1153 00:54:02.890 --> 00:54:05.619 Courant Events Right: The question is, how does this. 1154 00:54:07.230 --> 00:54:10.919 Courant Events Right: Where would this appear in the language of nonlinear sigma models? 1155 00:54:11.360 --> 00:54:12.780 Christophe Raymond Garban: So, I think… 1156 00:54:12.780 --> 00:54:20.260 Courant Events Right: The first perturbation on which we don't have a result, and we believe this would go, like Paul Yakov, to exponentially get all beta, this would be 1157 00:54:20.970 --> 00:54:24.060 Courant Events Right: This would fit well within nonlinear Sigma. 1158 00:54:24.060 --> 00:54:26.480 Christophe Raymond Garban: Right. 1159 00:54:26.480 --> 00:54:31.140 Courant Events Right: Here, I don't change the metric, I change the scalar product. 1160 00:54:31.140 --> 00:54:33.610 Christophe Raymond Garban: It is also detected in the potential term. 1161 00:54:33.630 --> 00:54:40.600 Courant Events Right: Yes, so it's more like adding a small potential term. One way to see the difference is that a nonlinear sigma model. 1162 00:54:40.850 --> 00:54:41.790 Christophe Raymond Garban: The grown stage… 1163 00:54:41.790 --> 00:54:44.630 Courant Events Right: It could be in any directions. 1164 00:54:45.170 --> 00:54:47.500 Christophe Raymond Garban: Reggie's community, though. 1165 00:54:48.860 --> 00:54:52.749 Christophe Raymond Garban: Well, so let me, let me go… 1166 00:54:52.750 --> 00:54:56.869 Courant Events Right: here, if I take this perturbed, what is it? 1167 00:54:57.880 --> 00:55:00.870 Courant Events Right: This… this first perturbation here. 1168 00:55:00.980 --> 00:55:07.709 Courant Events Right: Here, even if I have this flat domain, if I don't talk about entropy, if I take any constant field. 1169 00:55:08.360 --> 00:55:09.270 Christophe Raymond Garban: I pay… 1170 00:55:09.270 --> 00:55:11.050 Courant Events Right: exponential zero. So, in that. 1171 00:55:11.050 --> 00:55:11.730 Christophe Raymond Garban: zones? 1172 00:55:11.730 --> 00:55:17.569 Courant Events Right: These perturbations will have more ground states than the one we consider with modifying scalar products. 1173 00:55:17.730 --> 00:55:19.640 Courant Events Right: Because this second perturbation. 1174 00:55:19.870 --> 00:55:20.979 Christophe Raymond Garban: Only the points. 1175 00:55:20.980 --> 00:55:22.880 Courant Events Right: In the equator will be… 1176 00:55:22.880 --> 00:55:24.160 Christophe Raymond Garban: of human departments. 1177 00:55:24.160 --> 00:55:24.630 Courant Events Right: Yep. 1178 00:55:24.630 --> 00:55:25.660 Christophe Raymond Garban: just… 1179 00:55:25.660 --> 00:55:29.609 Courant Events Right: to look at the RG implications involved in additional companies. 1180 00:55:34.450 --> 00:55:39.580 Courant Events Right: And I think that's what Slava was doing on this piece of paper in IHS. 1181 00:55:39.920 --> 00:55:42.030 Courant Events Right: It wasn't… you didn't… 1182 00:55:44.160 --> 00:55:44.730 Christophe Raymond Garban: reading. 1183 00:55:50.700 --> 00:55:51.949 Christophe Raymond Garban: Okay, thank you. 1184 00:55:58.680 --> 00:55:59.290 Christophe Raymond Garban: Can you… 1185 00:55:59.290 --> 00:56:00.060 Courant Events Right: Hold on. 1186 00:56:00.860 --> 00:56:01.630 Christophe Raymond Garban: What does Bill? 1187 00:56:01.630 --> 00:56:03.139 Courant Events Right: about parish people. 1188 00:56:04.800 --> 00:56:05.840 Courant Events Right: You won't… 1189 00:56:06.470 --> 00:56:07.259 Christophe Raymond Garban: from getting your… 1190 00:56:07.260 --> 00:56:07.730 Courant Events Right: recognized. 1191 00:56:08.020 --> 00:56:09.650 Christophe Raymond Garban: I agree, control. 1192 00:56:09.790 --> 00:56:20.200 Courant Events Right: is large, and… You mean what would be known regularly about implementing. 1193 00:56:20.200 --> 00:56:22.480 Christophe Raymond Garban: So… 1194 00:56:22.480 --> 00:56:26.130 Courant Events Right: They managed to… 1195 00:56:28.630 --> 00:56:29.220 Christophe Raymond Garban: over the stretch. 1196 00:56:29.220 --> 00:56:30.950 Courant Events Right: over the implementation of our. 1197 00:56:31.780 --> 00:56:35.440 Christophe Raymond Garban: So, I think maybe I'm moving one first. 1198 00:56:35.440 --> 00:56:36.190 Courant Events Right: difficulty. 1199 00:56:36.190 --> 00:56:36.680 Christophe Raymond Garban: is… 1200 00:56:36.680 --> 00:56:45.690 Courant Events Right: It's a background field method to even define precisely what would be this low mode background field around which we are trying to. 1201 00:56:45.690 --> 00:56:46.540 Christophe Raymond Garban: being spent. 1202 00:56:46.660 --> 00:56:52.140 Courant Events Right: So I think this is… But I'm pretty sure I've seen some papers in the view, some papers of the character and the. 1203 00:56:52.760 --> 00:56:53.470 Christophe Raymond Garban: You can easily. 1204 00:56:53.470 --> 00:56:55.270 Courant Events Right: We're all about being open. 1205 00:56:55.490 --> 00:56:56.459 Christophe Raymond Garban: It doesn't go as well. 1206 00:56:56.460 --> 00:56:57.090 Courant Events Right: savings. 1207 00:56:57.090 --> 00:57:02.500 Christophe Raymond Garban: So, Balaband, I have never read Balaband. 1208 00:57:02.500 --> 00:57:02.850 Courant Events Right: video. 1209 00:57:02.850 --> 00:57:04.680 Christophe Raymond Garban: But… 1210 00:57:04.680 --> 00:57:05.690 Courant Events Right: Maybe, 1211 00:57:05.690 --> 00:57:06.310 Christophe Raymond Garban: Indeed. 1212 00:57:06.310 --> 00:57:07.650 Courant Events Right: he implemented the. 1213 00:57:07.650 --> 00:57:10.240 Christophe Raymond Garban: RG flow, which gives lung. 1214 00:57:10.240 --> 00:57:10.850 Courant Events Right: and all the. 1215 00:57:10.850 --> 00:57:11.250 Christophe Raymond Garban: was. 1216 00:57:11.250 --> 00:57:14.980 Courant Events Right: such nonlinear sigma model when you are in dimension 3. 1217 00:57:15.560 --> 00:57:17.010 Christophe Raymond Garban: So, it's a… 1218 00:57:17.010 --> 00:57:21.449 Courant Events Right: It's an implementation of RGflow to detect that, indeed, you do bring. 1219 00:57:21.450 --> 00:57:24.299 Christophe Raymond Garban: Ranking the symmetry in dimension 3. 1220 00:57:24.560 --> 00:57:26.450 Christophe Raymond Garban: Indeed, I've never read, actually. 1221 00:57:26.450 --> 00:57:28.489 Courant Events Right: I should have, it's a good question to ask. 1222 00:57:28.920 --> 00:57:29.630 Christophe Raymond Garban: what does. 1223 00:57:29.630 --> 00:57:30.179 Courant Events Right: his argument. 1224 00:57:30.180 --> 00:57:31.409 Christophe Raymond Garban: and two in dimension. 1225 00:57:31.410 --> 00:57:31.820 Courant Events Right: who… 1226 00:57:32.150 --> 00:57:34.180 Christophe Raymond Garban: Regarding this question here. 1227 00:57:35.490 --> 00:57:37.330 Christophe Raymond Garban: Being in New York, I would just know. 1228 00:57:37.330 --> 00:57:39.099 Courant Events Right: on the Ronald's office, and… 1229 00:57:41.930 --> 00:57:43.660 Christophe Raymond Garban: I don't think you would know more. 1230 00:57:45.050 --> 00:57:45.790 Christophe Raymond Garban: Yeah. 1231 00:57:46.880 --> 00:57:50.070 Christophe Raymond Garban: Yeah, it's kind of weird, such as your event. 1232 00:57:55.510 --> 00:57:56.530 Courant Events Right: Sure, I can say in the. 1233 00:57:56.530 --> 00:57:57.990 Christophe Raymond Garban: sense of progress. 1234 00:57:57.990 --> 00:57:58.470 Courant Events Right: Everyone. 1235 00:57:58.740 --> 00:58:00.550 Christophe Raymond Garban: We can talk later. 1236 00:58:01.690 --> 00:58:03.820 Christophe Raymond Garban: Dave, one last question. 1237 00:58:09.540 --> 00:58:10.280 Courant Events Right: We've been lawyers. 1238 00:58:12.730 --> 00:58:15.549 Christophe Raymond Garban: Yes, the question mark. 1239 00:58:15.550 --> 00:58:16.000 Courant Events Right: your head. 1240 00:58:16.000 --> 00:58:16.520 Christophe Raymond Garban: Emma? 1241 00:58:16.890 --> 00:58:18.150 Christophe Raymond Garban: The question is, what… 1242 00:58:18.150 --> 00:58:18.810 Courant Events Right: we do in CRM. 1243 00:58:18.810 --> 00:58:19.660 Christophe Raymond Garban: them too. 1244 00:58:20.240 --> 00:58:23.030 Courant Events Right: So, CRM2 is a… is a… 1245 00:58:24.010 --> 00:58:26.320 Christophe Raymond Garban: is, first… 1246 00:58:26.320 --> 00:58:35.959 Courant Events Right: We'll try to explain, as best as we can, what is this conjectured line between the exponential decay and the parallel decay, using our. 1247 00:58:35.960 --> 00:58:39.980 Christophe Raymond Garban: flow, which probably, follows your option, that there is a. 1248 00:58:39.980 --> 00:58:41.649 Courant Events Right: The potential, and there is the… 1249 00:58:42.280 --> 00:58:45.300 Christophe Raymond Garban: the classicalizing that type. 1250 00:58:45.310 --> 00:58:49.140 Courant Events Right: And the theorem is, in the case of a hemisphere where it is. 1251 00:58:49.140 --> 00:58:50.139 Christophe Raymond Garban: It's known that there is. 1252 00:58:50.140 --> 00:58:52.559 Courant Events Right: exponential decay at arbitrarily low temperatures. 1253 00:58:52.560 --> 00:58:54.060 Christophe Raymond Garban: Here we can see. 1254 00:58:54.060 --> 00:58:56.409 Courant Events Right: change the scale of product, and as we said. 1255 00:58:56.410 --> 00:58:57.359 Christophe Raymond Garban: And when the… 1256 00:58:57.360 --> 00:58:58.470 Courant Events Right: KT kicks in. 1257 00:58:59.170 --> 00:59:00.980 Christophe Raymond Garban: And your theorem is that. 1258 00:59:00.980 --> 00:59:01.610 Courant Events Right: DKT does. 1259 00:59:01.610 --> 00:59:02.690 Christophe Raymond Garban: Risk in advance. 1260 00:59:02.690 --> 00:59:03.890 Courant Events Right: low enough temperature. 1261 00:59:03.890 --> 00:59:06.529 Christophe Raymond Garban: theorem will never go to this RG curve. 1262 00:59:06.740 --> 00:59:08.169 Christophe Raymond Garban: But it will go all the way. 1263 00:59:08.170 --> 00:59:09.289 Courant Events Right: to lambda equals 1. 1264 00:59:09.640 --> 00:59:12.620 Christophe Raymond Garban: And the other side of the CRM is that. 1265 00:59:12.620 --> 00:59:15.280 Courant Events Right: We managed to extend the proof by. 1266 00:59:15.280 --> 00:59:18.330 Christophe Raymond Garban: Rorschmitt and Mutenzuelan, with. 1267 00:59:18.330 --> 00:59:19.100 Courant Events Right: scalar. 1268 00:59:19.100 --> 00:59:21.799 Christophe Raymond Garban: Product preservation, all the way to this. 1269 00:59:21.800 --> 00:59:22.320 Courant Events Right: curve. 1270 00:59:24.800 --> 00:59:29.739 Courant Events Right: Right, so those blockers are exponential decay is still here, still here, still here. 1271 00:59:30.080 --> 00:59:32.540 Christophe Raymond Garban: Then… then we don't know, and then. 1272 00:59:32.540 --> 00:59:33.980 Courant Events Right: Here, we know that there is parallel. 1273 00:59:34.140 --> 00:59:34.980 Christophe Raymond Garban: That's what… 1274 00:59:34.980 --> 00:59:36.040 Courant Events Right: with your MC. 1275 00:59:55.360 --> 00:59:56.709 Christophe Raymond Garban: You know, people. 1276 00:59:57.280 --> 00:59:57.990 Courant Events Right: Yeah. 1277 00:59:58.490 --> 00:59:59.610 Courant Events Right: That's the other…