WEBVTT 1 00:00:00.030 --> 00:00:01.020 Courant Events Right: Oh, who they? 2 00:00:15.360 --> 00:00:21.490 Courant Events Right: No, no, no, I'm over here. Okay, I'm just thinking my way over. 3 00:00:27.410 --> 00:00:28.120 Courant Events Right: What do you think? 4 00:00:28.480 --> 00:00:29.420 Courant Events Right: Pretty important. 5 00:00:50.360 --> 00:00:57.480 Courant Events Right: Congratulations. 6 00:00:57.580 --> 00:01:11.060 Courant Events Right: Okay, so, let's have the last talk of the day. So, Matt, we have the pleasure to have Nate from Jena, and then he will talk about the critical dynamic model. 7 00:01:11.340 --> 00:01:15.499 Courant Events Right: All right, thank you very much. Hi everyone, can you all hear me well? 8 00:01:16.040 --> 00:01:25.270 Courant Events Right: Okay, very nice. I'd like to start by thanking the organizers for their invitation. It's truly a pleasure to be here in front of such an impressive audience. 9 00:01:25.690 --> 00:01:35.550 Courant Events Right: So, I'll try to tell you, some, results. I think the title I was asked, to present was Massive SLE, and… 10 00:01:35.550 --> 00:01:47.969 Courant Events Right: Then I negotiated with him so that he could also include massive field theory, and, didn't tell the organizers that, actually what I will talk about is the uncritical diagram model. 11 00:01:47.990 --> 00:01:50.649 Courant Events Right: So I apologize in advance. 12 00:01:50.810 --> 00:02:08.480 Courant Events Right: Okay, so I, I, I tried to start with some, sort of generic overview, and, talking a bit about, criticality, conformal ambiance, and of criticality. So, I have in mind some, lattice model of, statistical mechanics, maybe easing. 13 00:02:09.020 --> 00:02:17.949 Courant Events Right: And as is well known, when we look at the critical points, so, you know, there is a parameter that we tune, let's say the temperature or inverse temperature. 14 00:02:18.000 --> 00:02:34.010 Courant Events Right: And, at… when we look at the critical point, what we see is the emergence of these, conformal symmetries, as is well known from this, paper of Polyakov Semologikov, which really gave rise to this field of conformal field theory, I guess. 15 00:02:34.420 --> 00:02:45.769 Courant Events Right: And as is well known, when we look away from the critical points, we typically see exponential decay of correlations, and this leads to basically trivial significance. 16 00:02:46.880 --> 00:02:48.460 Courant Events Right: So, 17 00:02:48.720 --> 00:02:59.000 Courant Events Right: The kind of topic today is what happens when we look not exactly at the critical point, but not really far away from it, so pretty close to the critical point. 18 00:02:59.250 --> 00:03:11.179 Courant Events Right: So you can think that, you know, this temperature, you don't take it exactly equal to the critical value, you take it close to it in a way that depends on the mesh size where your model is living. 19 00:03:11.600 --> 00:03:17.770 Courant Events Right: And when you do that, what you see is that you obtain scaling limits that are non-trivial. 20 00:03:17.980 --> 00:03:28.840 Courant Events Right: And, when you look at the scaling limits on infinities and maybe small scales, you see that they behave just like their critical cousins, so, conformally. 21 00:03:29.120 --> 00:03:39.660 Courant Events Right: But when you look at them on, you know, large scales, then you do observe this exponential decay of correlations. So they are sort of a way of interpolating between 22 00:03:39.660 --> 00:03:49.199 Courant Events Right: His conformal field theories, conformal invient scaling limits, and, the ones that are trivial and have exponential duplicate correlations. 23 00:03:49.310 --> 00:03:54.010 Courant Events Right: Which you get when you take the critic… the parameter to be different from the critical. 24 00:03:55.160 --> 00:04:07.899 Courant Events Right: So, for these, kind of near-critical scale mimics, you will have, in general, no conformal invariance, and this should lead to, connections with, sort of generic, quantum field fields. 25 00:04:09.350 --> 00:04:27.740 Courant Events Right: Now, you know, these lattice models of statistical mechanics, often you can view them in sort of two different ways. It's a question of choice, how you choose to view them. Either you can view them geometrically, because you're thinking that there is a random curve that describes interfaces. 26 00:04:27.750 --> 00:04:44.449 Courant Events Right: Or you can think of it as a random field, as a height function, maybe, or something like this. And depending on how you choose to think about this model, you know, these two points of view, they lead to sort of fairly different descriptions of these models. 27 00:04:44.600 --> 00:04:48.080 Courant Events Right: And I'll try to examine both of these points of use here. 28 00:04:50.130 --> 00:05:06.750 Courant Events Right: So, my own interest in this topic, I think, started when I read a paper of Makarov and Spirnov, which I think is very inspirational, and there is this paragraph here in this paper that I like very much. 29 00:05:06.750 --> 00:05:22.240 Courant Events Right: And I read, I will read it here. It says, the key property of SLE is its control maliance, which is expected in 2D lattice models only at predictability. And the question naturally arises, can SLE success be replicated for both critical models? 30 00:05:22.590 --> 00:05:35.380 Courant Events Right: In most of critical cases, to obtain a non-traditional scaling limit, one has to adjust some parameter, sending it at an appropriate speed to the critical value, and such limits lead to massive field series. 31 00:05:35.490 --> 00:05:47.049 Courant Events Right: So, this paper is, you know, has lots of, I think still today, many, many interesting questions, and I encourage people to… it's quite short. I encourage people to look at it. 32 00:05:48.520 --> 00:05:56.769 Courant Events Right: Right, so… So let me start with a geometrics perspective, perhaps. So, Markov and Snyder Noft. 33 00:05:56.800 --> 00:06:13.740 Courant Events Right: they developed a family of examples, which they call the massive SLE. I should say that, for the most part, there's no really general theory of these massive SLEs, at least as far as I'm aware. So they sort of really developed a number of examples. 34 00:06:13.920 --> 00:06:16.699 Courant Events Right: And they gave partial arguments. 35 00:06:16.810 --> 00:06:23.739 Courant Events Right: For convergence of some interface of natural discrete models to these massive facets. 36 00:06:27.010 --> 00:06:41.110 Courant Events Right: Now, you know, for the other perspective, the fear-theoretic perspective, I'll try to give some evidence that one should observe in the living some interesting model, which is called the Sein-Gorden model. 37 00:06:41.210 --> 00:06:53.220 Courant Events Right: And one thing that's remarkable about this Sandgordon model is that it's a quantum field theory which is not conformal, but it is still integral. There are still many things you can compute exactly. 38 00:06:55.830 --> 00:07:04.150 Courant Events Right: So I'll try to illustrate these two ideas, these two perspectives, with some recent results on, in particular, the near-critical dimer model. 39 00:07:07.020 --> 00:07:20.230 Courant Events Right: So really, the idea would be that when we're going to be considering this diagram model near its critical point, we'll be able to observe a connection both to massive SL2 and to this sandgordon model. 40 00:07:20.840 --> 00:07:39.989 Courant Events Right: And the appearance of this same garden model, I learned recently, was predicted at least by Luke Kanoff in 1997, in the context of the six vertex model. Probably it's the same thing as what I'm talking about here, I will check very carefully, and it may well be that there's even some older references that I'm not aware of. 41 00:07:41.810 --> 00:07:46.880 Courant Events Right: And so these results, they are… 42 00:07:47.030 --> 00:07:56.350 Courant Events Right: Jones works, respectively, with the Lady Huntman Sievitz for the connection to Massive S72, and for Scott Mason and Luke Art for the connection to Sam Gordon. 43 00:07:58.580 --> 00:08:11.600 Courant Events Right: But first, let me give you a bit of an overview. I'll start with some review. What is, in particular, massive loop raise one of the massive SAV. This is perhaps the case where it's maybe easiest to understand what are those massive SAVs. 44 00:08:12.120 --> 00:08:31.510 Courant Events Right: then a very brief review of some examples, and then I talk a bit about the DMR model, in particular the 10 billion setup, and some review of earlier results at criticality, and then I'll describe some of these results, talking about the near-critical model, some informal statement of results. 45 00:08:32.320 --> 00:08:48.610 Courant Events Right: I'll, it's, you know, there's kind of too much to talk about in one hour, so I think I'll try to focus more on the sign-government model than on massive SME. So I'll try to give some review about this, because it's a really non-trivial object to define, and it's a very interesting model, in my opinion. 46 00:08:48.770 --> 00:09:00.880 Courant Events Right: And in particular, on the Kolman transform, which is something that is very relevant and important for how we push things. And if there's time, I'll talk a bit about notions of massive problematic. 47 00:09:02.400 --> 00:09:05.450 Courant Events Right: And please feel free to interrupt to ask other questions. 48 00:09:07.050 --> 00:09:20.469 Courant Events Right: So, let's start with some background. I talk about a massive luperation work, which was discussed in this paper of Magarov and Spirnov, but also, it turns out a bit earlier in the paper about Kittle. 49 00:09:20.760 --> 00:09:33.989 Courant Events Right: And, so here I have in mind that, we are considering a random walk. So I put percent Ypsilon, so n is the time, you know, the number of steps of the walk. If silent denotes the lattice mesh size. 50 00:09:34.870 --> 00:09:47.060 Courant Events Right: And we have a domain omega, and omega psidon is a discretization of this, of this domain, and the work starts from some interior points at zero, and is killed when it leaves the moonula. 51 00:09:47.770 --> 00:09:53.519 Courant Events Right: And then, as I think everyone knows, as the mesh size is not delta, the flip side and tends to zero. 52 00:09:53.680 --> 00:10:03.209 Courant Events Right: If you speed up this random walk appropriately, I don't care how, but of course we know how, then, speeding up this random walk, yeah, then this converts to Browning motion. 53 00:10:04.280 --> 00:10:13.430 Courant Events Right: Now, I'm interested in the loop erasure, meaning the process that you obtain as you erase the loop chronologically as you make them. 54 00:10:14.280 --> 00:10:18.990 Courant Events Right: And when you do that, you obtain a simple curve, random simple curve. 55 00:10:19.140 --> 00:10:22.170 Courant Events Right: And, really one of the landmark results, 56 00:10:22.310 --> 00:10:27.960 Courant Events Right: SLE Theory, which is this fantastic work of Laura Schm and Gerner from 2002, 57 00:10:28.990 --> 00:10:39.340 Courant Events Right: Which shows that, as the mesh size tends to zero, the loop ratio of this random mark, so those phrase on the mock, converges to a certain random simple curve, which is called radial opacity. 58 00:10:40.720 --> 00:10:41.520 Courant Events Right: Okay. 59 00:10:43.540 --> 00:10:55.380 Courant Events Right: So, so far, so good. Now, what I'm interested is considering an off-pretty good observation. So, to do that, I'm gonna fix a number M in R, and this number, I think, valid as the mass. 60 00:10:56.170 --> 00:11:15.420 Courant Events Right: And what we are doing is, so I'm gonna have this, process XM epsilon, M, because there's a mass in this, in this problem. And, now, at each step, I suppose that, the walk, can be killed. And it's killed with a probability which is small. 61 00:11:15.590 --> 00:11:25.679 Courant Events Right: which is, just, M squared, epsilon squared over 2. So, you know, this probability tends to zero. It's, at each step, you've barely notice a difference with the critical model. 62 00:11:26.570 --> 00:11:34.699 Courant Events Right: So it sends it to zero, which is the critical value for this model, at an appropriate speed, just like, Michael and Stephen have said, they are 63 00:11:35.060 --> 00:11:36.260 Courant Events Right: description. 64 00:11:36.510 --> 00:11:42.479 Courant Events Right: And, so it's killed with this probability, and if it's not killed, it just continues on to a randomly chosen neighbor. 65 00:11:44.150 --> 00:11:47.580 Courant Events Right: So, so far it's just a work, but I'm going to loop erase it in a second. 66 00:11:48.220 --> 00:11:51.450 Courant Events Right: So is it clear? I just, you know, 67 00:11:51.600 --> 00:12:01.820 Courant Events Right: toss the point before moving, with some probability, I get steal, I get killed, and with some other probability, I continue, and if I continue, I just, follow on at number. 68 00:12:02.550 --> 00:12:05.039 Courant Events Right: It's one of my four possible menus. 69 00:12:05.600 --> 00:12:06.600 Courant Events Right: Okay. 70 00:12:07.050 --> 00:12:14.220 Courant Events Right: And, what I'm going to do is I'm going to condition this work to leave the domain omega without being killed. 71 00:12:14.470 --> 00:12:24.010 Courant Events Right: And, you know, this has the effect of making the path a bit shorter, I guess, right? So it doesn't want to hang around too much, because if it hangs around too much, it risks being killed. 72 00:12:24.120 --> 00:12:29.410 Courant Events Right: So that modifies, on a global scale, the law of the process. 73 00:12:31.080 --> 00:12:35.859 Courant Events Right: And then we may ask, what is the scaling limit of the loop measure of this process? 74 00:12:36.380 --> 00:12:40.730 Courant Events Right: So this is a question that was asked specifically in the paper of Makarov and Smyr. 75 00:12:42.330 --> 00:12:54.789 Courant Events Right: So you have, here a simulation. I don't know how well you can see these pictures. I can see them. Can you see them in the back? Oh, good. You can see, these two pictures, it's a bit like a before and after, you know, 76 00:12:55.260 --> 00:12:58.670 Courant Events Right: The type of feature that you see when you see the ads in the subway. 77 00:12:58.790 --> 00:13:18.560 Courant Events Right: So this is an ordinary luperation on a walk, and this is the after picture when we apply this massive procedure. And you can see that, you know, distance… well, on this picture at least, we get a slightly straighter pass, probably illustrating the fact that the walk itself is a bit shorter. 78 00:13:21.830 --> 00:13:41.619 Courant Events Right: Right, so, the theorem, which was stated, but not really proved in the paper of Makarov and Sirnov, and, I think also discussed implicitly, at least in the work of, Bawat and Kipolar, but really the details were given in a very nice paper of, Chad Park and Juan. 79 00:13:41.790 --> 00:13:57.809 Courant Events Right: shows that as epsilon tends to zero, the loop ratio of this massive work, conditioned to exit the domain without being healed, converges to a certain limit, which is this massive SLE2, as the match size epsilon cancels. 80 00:13:59.710 --> 00:14:06.340 Courant Events Right: And, this is a, you know, a random curve. It's absolutely continuous with respect to ordinary SLE tool. 81 00:14:06.580 --> 00:14:13.649 Courant Events Right: And this kind of absolute continuity is, in general, expected for massive staking limits corresponding to acidic carbon. 82 00:14:13.880 --> 00:14:17.240 Courant Events Right: Whenever the primary kappa is less or equal to 4. 83 00:14:18.510 --> 00:14:29.409 Courant Events Right: So the way this works, this massive SLE, camera, if one reads between the line, I think their definition in microphid spinoff is the following. 84 00:14:29.560 --> 00:14:43.829 Courant Events Right: if I pick a kappa, then I can associate to SLE kappa a certain Martingale observable. So something which will be a martingale as a function of time when I reveal more and more of the curve. 85 00:14:43.930 --> 00:14:59.459 Courant Events Right: And this marking it observable, you know, has the following form. It's a power ze to the beta times z prime to the sigma, where beta is a free parameter, and sigma is related to beta with a certain relation. 86 00:15:00.160 --> 00:15:07.779 Courant Events Right: And Z of is just a standard Levner floor. I don't want to get too much into the details of SLE, because it would take us too much time. 87 00:15:07.930 --> 00:15:10.739 Courant Events Right: So we get a certain marking dip, that's the point. 88 00:15:10.980 --> 00:15:23.559 Courant Events Right: And, there's a natural way to associate to each such multilingual observable a certain massive analog. It goes using the resolvent equation for autosphino. 89 00:15:24.120 --> 00:15:32.760 Courant Events Right: And by definition, we would say that the curve is a massive SLE if this massive observable indeed is a multiple. 90 00:15:33.670 --> 00:15:39.320 Courant Events Right: Interesting. Couldn't this also be done by just… The appropriate branding loop soup. 91 00:15:39.600 --> 00:15:59.450 Courant Events Right: Okay, there's a question from Greg, asking, could this be done directly using the wrong loop soup? And, probably, I'm guessing there's a possibility to do that. I'm not completely sure. I think, you know, it's not clear to me whether this definition is the best definition. 92 00:15:59.820 --> 00:16:15.500 Courant Events Right: Okay, okay. But at least it's what you find in the paper of blackout and scale up. As I said, you know, there's no real theory about this. For instance, I'm not sure the existence of a curve satisfying this probability is known, in general. 93 00:16:15.610 --> 00:16:17.360 Courant Events Right: So… 94 00:16:17.840 --> 00:16:21.870 Courant Events Right: Yes, maybe there… if there's a way to do it with a brand new, it probably would be quite interesting. 95 00:16:27.510 --> 00:16:31.340 Courant Events Right: Can one write an exact satisfaction for this, but again, right? 96 00:16:33.140 --> 00:16:34.529 Courant Events Right: I didn't catch them. 97 00:16:35.130 --> 00:16:38.970 Courant Events Right: I make a statement, but you can say that it's 18. 98 00:16:40.620 --> 00:16:42.869 Courant Events Right: That's okay. 99 00:16:43.200 --> 00:16:55.280 Courant Events Right: So the question is, can one write an exact instruction? It's a statement. And the answer is yes. Sorry, yes. 100 00:16:57.300 --> 00:17:12.170 Courant Events Right: Now, there's another question. Okay. Yeah. How does that relate to the mass? So, yeah, so, what happens is, so, often, these, 101 00:17:12.180 --> 00:17:19.709 Courant Events Right: Martingale observable. So, in principle, if I fix a silicappa for every guitar, this M is a Martingale. 102 00:17:19.780 --> 00:17:29.270 Courant Events Right: But often what happens is that there is one natural choice of beta, because you can prove for this value of beta, there is an analog discrete part of it. 103 00:17:29.940 --> 00:17:31.069 Courant Events Right: Basically, that's it. 104 00:17:31.500 --> 00:17:39.939 Courant Events Right: For every choice of beta, you have a multimed observable, and this multimed observable characterizes the locals say. 105 00:17:42.640 --> 00:17:46.800 Courant Events Right: And then the rule does not depend on that. It's always the lowest. That makes sense? 106 00:17:49.990 --> 00:18:08.879 Courant Events Right: So that's a bit the story for this massive SLE. So just very briefly, some, some examples. It's known that the harmonic Explorer converges to some, to a massive SLE4. This was, kind of sketched in this paper of Michael, and the 107 00:18:08.910 --> 00:18:12.529 Courant Events Right: Details were given by Leonie Papon fairly recently. 108 00:18:13.330 --> 00:18:26.800 Courant Events Right: It's also known that the lever lines of the massive Gaussian tree field are the same massive SL4, and they also coincide with a massive Brian and soup cluster outer boundary. It's also a result of Leonie. 109 00:18:28.030 --> 00:18:42.860 Courant Events Right: One can also consider a magnetically perturbed piecing model, and Leonie recently proved that the interfaces, the corresponding interfaces, they converge to a curve which is absolutely continuous with respect to SL3. 110 00:18:43.020 --> 00:18:52.710 Courant Events Right: And I guess one can think it's a form of massive SL3, but she did not prove that it coincides with the definition of fragdolfinorov, as I gave it. 111 00:18:53.170 --> 00:18:54.159 Courant Events Right: moments ago. 112 00:18:55.840 --> 00:19:14.589 Courant Events Right: So that's a very short review of what that means. Yes, magnetic field, exactly. So that's what I meant by that. The question was, did I mean magnetic field or temperature perturbation? You could do both for the ESIG model, but this result is about magnetic perturbation. 113 00:19:14.940 --> 00:19:16.359 Courant Events Right: You had a small mic. 114 00:19:20.040 --> 00:19:28.760 Courant Events Right: Okay, so I want to move to the Dimer model, because this is where I'll be able to say some novel results. 115 00:19:29.080 --> 00:19:34.590 Courant Events Right: And the driver model is a very nice model of statistical mechanics. 116 00:19:34.810 --> 00:19:40.249 Courant Events Right: Very briefly, it's a, you know, you have a planar graph, and you consider perfect matchings. 117 00:19:40.970 --> 00:19:47.280 Courant Events Right: on this graph, by which I mean a subset of edges which cover every vertex exactly. 118 00:19:48.970 --> 00:19:59.880 Courant Events Right: And it's well known that, I can reformulate… well, I shouldn't point, because different people cannot see. So it's well known that we can reformulate, 119 00:19:59.930 --> 00:20:09.980 Courant Events Right: a dimer configuration as a tiling of an associated region on the plane. So what we do is we imagine that every vertex is at the center of a face. 120 00:20:11.180 --> 00:20:21.630 Courant Events Right: And when I have a diamond edge, I just glue the two adjacent faces together, and that forms, in this case, the rhombus, the general gluing of two faces. 121 00:20:21.750 --> 00:20:26.370 Courant Events Right: And, that ties the associate division of the pay. 122 00:20:26.570 --> 00:20:31.979 Courant Events Right: And it's a model that is very well known, particularly because it has a rich behavior. 123 00:20:32.200 --> 00:20:43.569 Courant Events Right: It has a phase diagram with multiple phases. It can be either in the decree, the gas, or solid phases, and this corresponds to the beautiful work of Kenyon of Bugatt and Sheikh. 124 00:20:45.860 --> 00:21:02.160 Courant Events Right: So, I'm going to talk just about what is happening on the square lattice, and I want to give you a bit of… to spend a bit of time talking about the combinatorial details of the Daimler model, because unlike other models of statistical mechanics. 125 00:21:02.160 --> 00:21:10.310 Courant Events Right: The Dameron model is extremely sensitive to the exact microscopic detail, the boundary conditions, and things like that. 126 00:21:11.020 --> 00:21:16.029 Courant Events Right: So, what is a temporary domain? I'll start… 127 00:21:16.180 --> 00:21:26.489 Courant Events Right: By, by saying that, we're gonna do… we're gonna color every vertex of the square that is black and white in the usual check-born, check, check for the pattern. 128 00:21:27.340 --> 00:21:45.409 Courant Events Right: But in the case of the Dammer model, it actually pays off to subdivide further the black vertices into two classes. So, you know, the black vertices of the square lattice are themselves a copy of the square lattice, and I'm going to subdivide those into two in the checkbox pattern. 129 00:21:45.470 --> 00:21:51.340 Courant Events Right: Just as I did with the original database. So really, I have two types of black vertices, which have helpfully 130 00:21:51.420 --> 00:21:54.440 Courant Events Right: Decided to color blue and red. 131 00:21:54.660 --> 00:22:03.260 Courant Events Right: And so the ones that are not colored, which are actually black on the picture, but you should ignore that, they're what I think of as white vertices. 132 00:22:03.410 --> 00:22:10.089 Courant Events Right: Okay, so we have the black vertices, which are important, and then this can be blue or red, and we have the rest of the vertices, which are white. 133 00:22:11.990 --> 00:22:16.550 Courant Events Right: Okay. And then the 10th alien domain, 10 billion boundary condition. 134 00:22:16.550 --> 00:22:32.629 Courant Events Right: is saying that as we navigate along the boundary, only black vertices of one given type, let's say the blue type, are exposed. Okay, so you can see that on this domain, as I go along the boundary, I only see blue vertices. 135 00:22:32.860 --> 00:22:42.860 Courant Events Right: And, I have… if I just do that, it turns out this will not be tied above this domain. There will not be dynamic configuration associated with that. 136 00:22:43.000 --> 00:22:51.450 Courant Events Right: And, I have to remove the corner, and I pick one corner and just remove it. So here, I've chosen to remove the top right. 137 00:22:51.760 --> 00:22:54.820 Courant Events Right: corner. But you see that if I hadn't done that. 138 00:22:54.950 --> 00:23:03.389 Courant Events Right: If I hadn't removed this corner, then every black vertex on the boundary would be blue. Clear? So that's what I call a templeion domain. 139 00:23:03.590 --> 00:23:13.159 Courant Events Right: And it turns out this boundary condition might look a bit funny, but I guarantee you this is really the most canonical boundary condition one can put on the diner model. 140 00:23:13.720 --> 00:23:19.300 Courant Events Right: It's canonical from many different perspectives, and this is one where we can really say a lot of things about the background. 141 00:23:22.000 --> 00:23:36.170 Courant Events Right: Right, so I've told you what was the Daimler configuration, but I haven't really specified the load on dimer configuration, and here we're going to imagine that our graph comes with weights, and the dimer model is that we pick 142 00:23:36.170 --> 00:23:42.570 Courant Events Right: Diameter configuration, which is the probability proportional to the product of the edge weighted in the configuration. 143 00:23:44.380 --> 00:23:49.039 Courant Events Right: And then you normalize it with a factor 1 of the Z, just so that it's proper definition. 144 00:23:50.610 --> 00:24:01.770 Courant Events Right: And, there is this, when, you know, all the weights are chosen to be equal to 1, so that would be the critical case, then there is this really landmark result of a Rick Canyon. 145 00:24:01.970 --> 00:24:09.700 Courant Events Right: who proved that, the diamond model is a scientifically conformal invariant. I think this was the first proof of conformal invariance. 146 00:24:09.980 --> 00:24:11.210 Courant Events Right: Probability? 147 00:24:11.610 --> 00:24:16.220 Courant Events Right: And, you know, much more than that, also the introduction of the method. 148 00:24:16.740 --> 00:24:20.270 Courant Events Right: district, for the monkeys, so really a landmark developer. 149 00:24:20.890 --> 00:24:36.929 Courant Events Right: And what he proved really specifically is that there is an associated height function, which maybe I don't really define because it would take me too much time, but there is an associated height function which is associated to the Denver model. 150 00:24:37.270 --> 00:24:47.749 Courant Events Right: And his result really says that as the mesh size tends to zero, when we consider the diner model on 10 billion approximation of the Gilon domain. 151 00:24:48.130 --> 00:24:58.399 Courant Events Right: Then we… the height function converges to a multiple, 1 over square root phi times that, times the Gaussian pre-field with the… with Dirichlet boundary conditions. 152 00:24:59.550 --> 00:25:07.020 Courant Events Right: And the fact that Jewish-led boundary conditions occur is partly related to the fact that we have chosen Tempelian boundary conditions. 153 00:25:08.000 --> 00:25:09.460 Courant Events Right: So far, so good. 154 00:25:10.520 --> 00:25:18.550 Courant Events Right: So this is the critical case. This is, here we've chosen all the weights equal to 1, and we have a conformantiance. 155 00:25:19.570 --> 00:25:20.480 Courant Events Right: So… 156 00:25:20.990 --> 00:25:40.590 Courant Events Right: Yeah, as I said, his result was proved first on the square lattice, and then the increase that he had extended it to full plane, with, you know, iso-rendel superposition, then zombion, really, on domains, assuming, that these domains have a straight edge portion, and, there's some more recent results by myself and 157 00:25:43.790 --> 00:25:53.370 Courant Events Right: So, let me tell you some of the recent results that we have for dimers when we consider near-critical perturbation. 158 00:25:54.470 --> 00:26:07.109 Courant Events Right: So the near-critical model, was really introduced by Sunil Chita in 2012, and he introduced it on the square lattice, on the role plane, for what will correspond to the case, 159 00:26:08.690 --> 00:26:15.469 Courant Events Right: Later, we'll have a more general model, and this really turns out to be the best where alpha is concerned. 160 00:26:15.710 --> 00:26:18.490 Courant Events Right: And what he, what he showed was, 161 00:26:18.790 --> 00:26:26.039 Courant Events Right: You know, there is still a height function, and there is still a steady limit of this height function, but in this case, the limit is non-Gaussian. 162 00:26:27.650 --> 00:26:44.150 Courant Events Right: And, okay, so let me tell you, what is the kind of near political perturbation I'm doing. The basic parameter for this deformation will be a vector field. So alpha is going to be a function from the domain onto R2, 163 00:26:44.190 --> 00:26:51.880 Courant Events Right: Or the complex plane, depending on how you'd like to think about it. So vector field, I think of it as an external electromagnetic field. 164 00:26:53.090 --> 00:26:59.080 Courant Events Right: But I'm always going to assume that phr der derives from a certain potential, which I'm going to call V. 165 00:27:00.250 --> 00:27:07.010 Courant Events Right: So here's, an illustration. So you can think the domain is a square, and here's your, 166 00:27:07.110 --> 00:27:24.139 Courant Events Right: electromagnetic field alpha. And, let me tell you how we, define, associated, you know, once I specify this vector field for you, how do I specify a near political dynamic model? So I'm just going to change, the weights, they're not going to be equal to 1. 167 00:27:24.770 --> 00:27:30.900 Courant Events Right: And I'm gonna have… I'm gonna fix any black vertex Z, but a blue-black vertex. 168 00:27:30.930 --> 00:27:46.020 Courant Events Right: And, around each… I'm going to modify the edge weights around this vertex Z in the following manner. I'm going to take it to be exponential of the difference of the potential along the edge ZW. 169 00:27:46.780 --> 00:28:02.949 Courant Events Right: So, I should clarify that I have in mind the situation where we have a very small mesh size, so Z and W are really very close to one another, so V of Z and V of W are pretty close to one another, and this weight is pretty close to 1, because V is continuous. 170 00:28:04.350 --> 00:28:07.700 Courant Events Right: Okay, is that clear? How do I define that? 171 00:28:08.760 --> 00:28:17.479 Courant Events Right: So, around every black vertex which is blue, I will modify the edge weights in this manner, and otherwise, the edge weights remain one. 172 00:28:19.840 --> 00:28:28.530 Courant Events Right: So these edgeways, they are all unoriented, but I can think also that they define oriented ways on the lattice gamma, on the blue lattice. 173 00:28:28.710 --> 00:28:41.130 Courant Events Right: And, if I do a random walk, which is oriented, so, you know, I use C of Z and W as the weight going from Z to its blue neighbor in the direction W. 174 00:28:42.430 --> 00:28:58.070 Courant Events Right: And if I do a random walk with this weight, it turns out the work on this blue lattice would converge to the launch band diffusion, meaning dx source DX equals D of a random motion, plus a drift given by this, 175 00:28:58.990 --> 00:29:00.000 Courant Events Right: gradient. 176 00:29:01.180 --> 00:29:02.000 Courant Events Right: Oh. 177 00:29:02.460 --> 00:29:11.179 Courant Events Right: So this is… if you want to understand the meaning of these weights, that's what they do. So here is a simulation, which is… 178 00:29:11.520 --> 00:29:21.609 Courant Events Right: I think you can probably tell that I'm cheating. So, yes, this is the one simulation due to me, 179 00:29:21.720 --> 00:29:22.440 Courant Events Right: Still. 180 00:29:23.090 --> 00:29:29.690 Courant Events Right: So you imagine that you have this, you know, drift, and the drift is kind of pushing your wall towards the boundary. 181 00:29:31.190 --> 00:29:32.010 Courant Events Right: Okay. 182 00:29:32.200 --> 00:29:41.800 Courant Events Right: So, yes, yes, yeah, here. But I'm sure 183 00:29:42.450 --> 00:29:49.179 Courant Events Right: Yes, it's going to be very important that the drift derives my potential for what I'm going to say. 184 00:29:49.460 --> 00:29:57.009 Courant Events Right: Like, that plays a crucial role. There will be another assumption that the potential is going to be blocked X. This, I'm less certain whether it plays a role. 185 00:29:57.420 --> 00:30:02.049 Courant Events Right: But the fact that the rest of the natural is crucial for what we need to say. 186 00:30:02.280 --> 00:30:06.270 Courant Events Right: And I have no idea. 187 00:30:06.790 --> 00:30:10.530 Courant Events Right: I really need a few. I don't venture in this. 188 00:30:12.910 --> 00:30:22.679 Courant Events Right: Yes, there's a question. Also, the solution of the easy model, there is a mapping in the dimer point for points, I mean, you… 189 00:30:23.040 --> 00:30:42.780 Courant Events Right: Okay, the question is, there is… there are… there exists exact mapping between dimers and easing. Massive. Massive, exactly. So why don't I just start working with… with a easing model? This is a very good question, and it has to do with the fact that this mapping 190 00:30:42.780 --> 00:30:46.060 Courant Events Right: Don't behave well at all at boundary conditions. 191 00:30:46.640 --> 00:30:56.909 Courant Events Right: That's what you mean. They don't like to book the four partners. 192 00:30:57.120 --> 00:31:00.720 Courant Events Right: I'm not sure, I'm not sure this… Okay, okay, I'm not sure. 193 00:31:01.750 --> 00:31:14.190 Courant Events Right: Was there one more question? No, there is a paper by Julien Dubedin. There is a paper by Julien Dubedardin, which is called, bosonization, I guess, exact bosonization for these models, something like that. 194 00:31:14.330 --> 00:31:29.650 Courant Events Right: where this kind of relations are described. I don't know whether he specifically discussed the massive case, but probably one can do something in the massive case, but I think it behaves very badly at the boundary, this kind of excitement. 195 00:31:31.250 --> 00:31:31.940 Courant Events Right: Okay. 196 00:31:33.760 --> 00:31:48.330 Courant Events Right: Okay, so let me first tell you about the connection to massive SLE for this near-critical dynamo model. It turns out that… so first of all, I'm going to assume, as I said, my basic parameter 197 00:31:48.500 --> 00:32:02.430 Courant Events Right: to describe the deformation of the dynamic model is this vector field alpha, so I'm going to assume that alpha describes, you know, derives from the potential. Every time I specify alpha, I've specified for you a near-critical dynamer model. 198 00:32:02.590 --> 00:32:14.229 Courant Events Right: And I'm going to assume not only that my drift derives from a potential, but this potential is log convex, meaning that Laplacian P plus radian P squared is now negative over the R. 199 00:32:14.560 --> 00:32:15.890 Courant Events Right: And, yep. 200 00:32:16.240 --> 00:32:19.140 Courant Events Right: This is an option, an assumption that I'll come back to that in a second. 201 00:32:19.630 --> 00:32:31.080 Courant Events Right: So, the theorem, I mean, it's a bit hard to describe here because I don't have that much time. It turns out there are paths one can naturally associate it to the Dimer model. 202 00:32:31.420 --> 00:32:39.769 Courant Events Right: For those of you who know, there is a bijection called temperless bijection, which transforms the diamond model into spying trees. 203 00:32:39.920 --> 00:32:50.609 Courant Events Right: And, one can then naturally consider the spanning… the path connecting a point in the interior, to the boundary via the spanning tree. 204 00:32:51.010 --> 00:33:04.850 Courant Events Right: This is what this picture is showing you. And the theorem that we proved with Levi on Trumcippies a couple of years ago, well, a few years ago, is that, the scaling limit of this path is Makarov and Smirnov's mass infestibut. 205 00:33:05.480 --> 00:33:18.480 Courant Events Right: So I should say that this is what you get when the drift alpha is constant, it's exactly the process considered by Michael, as well as Chuck and M1. And when… 206 00:33:18.670 --> 00:33:25.210 Courant Events Right: Alpha is not a constant vector field, then the scaling limit is a form of mass either 2, which is normal. 207 00:33:27.580 --> 00:33:39.809 Courant Events Right: And, it turns out that, from this, it follows that, you know, if I consider the exit the associated height function, then this height function has, a scaling limit. 208 00:33:41.570 --> 00:33:49.860 Courant Events Right: And, we are also able to prove, at that time that the limiting height function satisfies the home format co-bargets. 209 00:33:50.970 --> 00:33:57.480 Courant Events Right: Now, we were also able to conjecture a precise link to the same golden model, and that's what I want to tell you next. 210 00:33:58.830 --> 00:34:01.020 Courant Events Right: Let's set up questions, is it? Yes. 211 00:34:01.180 --> 00:34:03.620 Courant Events Right: performance diet for the best sleep. 212 00:34:05.790 --> 00:34:11.150 Courant Events Right: The question is, do I have conformal invariance for the massive? So, the fact that it's massive. 213 00:34:11.340 --> 00:34:16.499 Courant Events Right: means exactly that I don't have conformal invariance, but I have conformal covalent. 214 00:34:17.080 --> 00:34:23.250 Courant Events Right: the, the, the massive T translates, so… But then the math become… and uniform. 215 00:34:23.699 --> 00:34:33.599 Courant Events Right: Yes, exactly. So, so thank you. The question is, but then the mask, is not… does not remain uniform, and that's, so thank you, that's a very good point. 216 00:34:33.600 --> 00:34:49.070 Courant Events Right: So for us, it's very important that we're able to work with masses that are viable, and indeed, drilled backfields that vary from base to base, because this is what, if you want to play with conformational variants, you will naturally be lens. 217 00:34:49.489 --> 00:34:55.620 Courant Events Right: who uses the terminology. Any massive fish theory will be conformity. 218 00:34:56.040 --> 00:35:04.809 Courant Events Right: Any message with Siri would be conformity? I don't know, I'm not, I don't know enough. 219 00:35:04.980 --> 00:35:08.140 Courant Events Right: Varying from point to point, 220 00:35:08.320 --> 00:35:13.549 Courant Events Right: I don't know how much transformation, but it sounds like you say, well, it's definitely the case. 221 00:35:13.720 --> 00:35:20.340 Courant Events Right: The comment is that, probably, in this sense, every conform… every quantum field theory is conformed guideline. 222 00:35:20.740 --> 00:35:22.059 Courant Events Right: Maybe quite a bit. 223 00:35:22.410 --> 00:35:26.430 Courant Events Right: Just a trade-off between this and the factory. 224 00:35:26.540 --> 00:35:28.139 Courant Events Right: Secure the fight for the prevention. 225 00:35:28.260 --> 00:35:40.850 Courant Events Right: You just wait. You wait a couple of months as you feel better. 226 00:35:41.000 --> 00:35:45.860 Courant Events Right: There's an interesting discussion. I will not refuse all of the discussion. 227 00:35:46.170 --> 00:36:04.979 Courant Events Right: So, let me, tell you a bit about this connection to San Gordon. Again, I'm going to assume that the drift factor in alpha derives from a potential which is not convex, and so, as I said, this assumption is that the Labashian plus gradient E squared is non-negative, and this function really plays the role of the mass. 228 00:36:05.600 --> 00:36:08.100 Courant Events Right: It turns out its assumptions come from a lot. 229 00:36:08.510 --> 00:36:13.679 Courant Events Right: And if I map 3 by conform and map, turns out it remains index. 230 00:36:15.140 --> 00:36:28.859 Courant Events Right: So, the, the theorem that is novel, you can find it on the archive from maybe a month or two ago, says that, so it proves this conjecture that we had formulated precisely in this newspaper research. 231 00:36:29.430 --> 00:36:30.940 Courant Events Right: living on from the cities. 232 00:36:32.200 --> 00:36:43.980 Courant Events Right: And, so we're going to assume that the domain contains a straight edge portion, or is the image of such a domain by a map, which is analytic in a neighborhood of Oliver, so pretty smooth. 233 00:36:44.940 --> 00:36:54.650 Courant Events Right: We're going to consider the standardized function associated to the near political dynamic model defined by alpha, defined by this vector. 234 00:36:55.090 --> 00:36:58.550 Courant Events Right: And the theorem says that as if sine tends to zero. 235 00:36:58.770 --> 00:37:10.780 Courant Events Right: the strike function converges to a certain field in low, in the sense that when I test it against the test function, it converges, converges low for arbitrary testing. 236 00:37:11.930 --> 00:37:17.270 Courant Events Right: It's one part of the human. Furthermore, so furthermore. 237 00:37:17.430 --> 00:37:25.339 Courant Events Right: Every time I pick a vector S1 through Sn of zeros and ones, think of them as signs. 238 00:37:25.500 --> 00:37:39.739 Courant Events Right: And every time I consider pairwise disjoint points, Z12Z and N omega, if I start considering the products of the polymorphic and anti-polymorphic derivatives. 239 00:37:39.740 --> 00:37:46.209 Courant Events Right: of the fields at the point Z1 through ZN, so the SI are telling you whether I consider the order method 240 00:37:46.510 --> 00:37:48.780 Courant Events Right: Or antagonic variables. 241 00:37:49.950 --> 00:37:55.369 Courant Events Right: So we can compute, these, correlation functions in recent sense, at least. 242 00:37:55.500 --> 00:37:58.239 Courant Events Right: And the theorem says that, 243 00:37:58.350 --> 00:38:06.400 Courant Events Right: Derivatives are given by, well, the correlation of the derivatives are given by a certain determinant of the matrix. 244 00:38:06.620 --> 00:38:13.449 Courant Events Right: Where, so, as I said, the DSI stands for Vietnachine derivative, so polymorphic or non-diolomorphic. 245 00:38:13.970 --> 00:38:19.210 Courant Events Right: And these, F, functions, there are certain, massive polymorphic functions. 246 00:38:19.420 --> 00:38:25.690 Courant Events Right: I'll say later what that means. And they satisfy a certain boundary condition, which is called square root of boundary. 247 00:38:27.070 --> 00:38:32.769 Courant Events Right: If you want to know what I mean by, in the weak sense, I mean, that's, 248 00:38:33.310 --> 00:38:46.890 Courant Events Right: yeah, this is missing some, some derivatives. I mean that, No, no, no, it's okay. So this is, this is concretely what it means, to get to, that's the, the, the, you know, the… 249 00:38:47.460 --> 00:38:53.969 Courant Events Right: the coalitions of the derivatives of the field are given by this determinant in a weak sense. 250 00:38:55.600 --> 00:38:56.410 Courant Events Right: Okay. 251 00:38:56.640 --> 00:39:03.689 Courant Events Right: So… so it's a certain, expression, determined expression for the correlations of the derivatives of the… 252 00:39:05.960 --> 00:39:08.399 Courant Events Right: No, you would get lost with his own. 253 00:39:10.840 --> 00:39:16.040 Courant Events Right: The question is, will I tell you what is F? And, it depends whether I have time or not. 254 00:39:16.860 --> 00:39:17.700 Courant Events Right: Maggie. 255 00:39:18.630 --> 00:39:21.080 Courant Events Right: You should behave well. 256 00:39:21.480 --> 00:39:29.019 Courant Events Right: Okay, so what about the connection to Sun Garden? So, I'm going to suppose omega is the hidden disk. 257 00:39:29.150 --> 00:39:35.150 Courant Events Right: And, alpha, I will have to impose some additional assumptions, which I won't, explain now. 258 00:39:35.950 --> 00:39:47.399 Courant Events Right: And, the theorem says that, in that case, we get a more precise expression, for the correlation of the polymorphic and anti-polymorphic derivatives in the field. 259 00:39:47.420 --> 00:39:57.970 Courant Events Right: And, it's given specifically by a multiple of, the same, correlation function for what's called the same modern model. 260 00:39:58.000 --> 00:40:04.500 Courant Events Right: He's a… mass profile row, and row is related to alpha in some very common. 261 00:40:05.770 --> 00:40:07.449 Courant Events Right: Gary the definition if you want. 262 00:40:08.270 --> 00:40:16.370 Courant Events Right: So, I haven't told you what is the expectation for San Gordon, and I need you to do that now. That's what I'm going to do now. 263 00:40:19.050 --> 00:40:21.540 Courant Events Right: There is a moment in the interest of time. 264 00:40:21.700 --> 00:40:23.019 Courant Events Right: Of course. Don't you? 265 00:40:24.070 --> 00:40:38.390 Courant Events Right: So, what is the Sang-Golden model? So, informally, I'm going to… the Sngoordan model with profile row, so I'm going to pick a function from omega to R now, it's a real value, function, and 266 00:40:38.530 --> 00:40:48.419 Courant Events Right: informally, the same Gordon low, needs, basically the low of the Gaussian free field, so that's this key GFF, sharp here. 267 00:40:49.570 --> 00:40:54.550 Courant Events Right: But it's really… it's reweighted by a certain, factor. 268 00:40:55.490 --> 00:40:59.709 Courant Events Right: Which is the exponential of a constant, times integral of the domain. 269 00:41:00.420 --> 00:41:06.250 Courant Events Right: of cosine of square root beta times the field times the mass profile rho Z. 270 00:41:08.000 --> 00:41:23.679 Courant Events Right: Okay, so, my, my, you know, one has to pay attention to, normalization conventions, because they are important here. There are different ways of normalizing the Gaussian-free field, corresponding to different ways of normalizing the Laplacian or the green function. 271 00:41:24.030 --> 00:41:33.659 Courant Events Right: And, here my convention is that the two-point function of this field is 1 over 2 pi times the log, plus, plus a downward term. 272 00:41:35.290 --> 00:41:35.990 Courant Events Right: Okay. 273 00:41:36.500 --> 00:41:38.980 Courant Events Right: Okay, so… 274 00:41:39.140 --> 00:41:47.960 Courant Events Right: Now, this was an informal definition, and I have to warn you right away that assigning a rigorous meaning to this expression is really far from straightforward. 275 00:41:48.350 --> 00:42:01.840 Courant Events Right: And, in fact, as beta increases, so beta was, you know, appeared in the definition, the model goes through a sequence of thresholds where the definition of the same garden becomes increasingly more difficult. 276 00:42:02.700 --> 00:42:13.210 Courant Events Right: And the first threshold is at beta equals 4 pi, which is known as the three Fallman point, and these are the thresholds they accumulate at beta equals 8 pi, who is a normalization. 277 00:42:14.660 --> 00:42:27.579 Courant Events Right: Now, 4 beta less than 4 pi, it turns out one can directly assign meaning to the integral that appeared in the exponential in front of the law of the Gaussian free slip, rather than putting derivatives. 278 00:42:27.830 --> 00:42:42.750 Courant Events Right: And one has to regularize the field, you know, it doesn't really make sense to take the cosine of the Gaussian free field, because the Gaussian free field is not pointwise defined, so the cosine is not directly defined. But I can regularize the field at scale, sine up. 279 00:42:42.940 --> 00:42:55.740 Courant Events Right: And consider the cosine of that, and renormalize appropriately, so it turns out one has to renormalize by epsilon to the minus beta over 4 pi. 280 00:42:56.820 --> 00:43:05.050 Courant Events Right: And as you said, epsilon tends to zero, it turns out this limit exists. It's known as imaginary mathematic arrows. 281 00:43:05.190 --> 00:43:10.359 Courant Events Right: And, you can look at the work of, Unilang, who's here, and the Saxman. 282 00:43:11.680 --> 00:43:19.470 Courant Events Right: So for beta less than 4 pi, one can directly assign a meaning to the Sandordan model using this definition that I'm explaining here. 283 00:43:20.990 --> 00:43:30.209 Courant Events Right: So, you know, that means that the same garden model associated with keta lessen for pi is well-defined, and it turns out, absolutely continuous with respect to the law of the gas approach. 284 00:43:31.330 --> 00:43:38.170 Courant Events Right: Now, at beta equals 4 pi, this integral ceases to make sense as a function of phi. 285 00:43:38.530 --> 00:43:47.379 Courant Events Right: Nevertheless, it turns out, still possible to make sense of this low, but then, presumably, that low is not absolutely continuous with respect to the gas flow. 286 00:43:50.100 --> 00:44:00.929 Courant Events Right: Okay, so I will explain later how to do that rigorously, but, so that's not all. Informally, at this free Fermion point, the sand garden model has additional symmetries. 287 00:44:01.080 --> 00:44:13.049 Courant Events Right: Which are known as Kohlman's correspondence. So, this Coleman correspondence, says that the Sang-Gordon model, with mass profile rho, is equivalent 288 00:44:13.200 --> 00:44:17.420 Courant Events Right: to a field of massive tree inferments. So. 289 00:44:17.830 --> 00:44:19.780 Courant Events Right: I tried to explain what that is. 290 00:44:19.960 --> 00:44:34.850 Courant Events Right: So basically, you should think that, in this massive reframing model, we have, at every point Z of the domain, we have a vector of psiZ, which consists of two components, psi 1 of Z, psi2 of Z, Charles Grassman variables. 291 00:44:34.930 --> 00:44:50.249 Courant Events Right: So these are what's called, Dirac Fermions, and in this model, massive free Fermion model, so this is something which has formal density given by the exponential of, this, action, this integral. 292 00:44:51.300 --> 00:44:56.220 Courant Events Right: And, S of psi, this action is, given in terms of, 293 00:44:56.440 --> 00:44:59.380 Courant Events Right: Jira footprint, like the massive Jira footprint. 294 00:45:01.310 --> 00:45:16.739 Courant Events Right: Okay, so this is the Kolman correspondence, and so as I said, it relates the same Golden model to the massive-free Fermion. Let me take a little bit of time to say what that means. First of all, you should think of this as a massive extension of the Boson-Fermion correspondence. 295 00:45:16.740 --> 00:45:22.639 Courant Events Right: What I mean by that is that if one takes rho equals zero, so that means no deformation. 296 00:45:22.640 --> 00:45:34.209 Courant Events Right: then this correspondence tells you that the Gaussian free field is equivalent to this free Ferment model, where the density, the formal density would be exponential of minus 297 00:45:34.290 --> 00:45:35.419 Courant Events Right: Isn't the goal? 298 00:45:36.700 --> 00:45:53.949 Courant Events Right: So that's, both of the main correspondence. And, one thing that I found really remarkable about this, command correspondence is that, the Sandgarden model, which is, you know, this sort of fairly complicated object, which has a load that turns out is really quite hard to define. 299 00:45:54.410 --> 00:46:07.469 Courant Events Right: on the probabilistic side. Well, when you look at it on this fermionic side, it kind of becomes trivial, because you get this, you get this expression for the formal density, and this is a quadratic expression. 300 00:46:07.470 --> 00:46:15.189 Courant Events Right: Which means, you know, it really is a free Fermion model. It's as if we were considering the massive gas and free field, but for Fermions. 301 00:46:16.980 --> 00:46:21.169 Courant Events Right: So it's like it becomes Gaussian, if you like, when you look at it on the Ferminal side. 302 00:46:21.400 --> 00:46:40.340 Courant Events Right: So, it sort of trivializes the same garden model, which I find quite remarkable. So, now, constructing the right-hand side rigorously is something that is relatively standard as far as understand mathematical physics. For instance, one can look at this work, Ben Flatto, Falco, and Metropolis. 303 00:46:40.430 --> 00:46:41.759 Courant Events Right: That's what people see. 304 00:46:41.950 --> 00:46:50.110 Courant Events Right: And this is a particular case of the theory model, which in general even allows for quartic interaction, a bit like in the 5-4 model, for those of you who know. 305 00:46:52.890 --> 00:46:53.770 Courant Events Right: Okay. 306 00:46:53.970 --> 00:47:06.750 Courant Events Right: So, now, the Kohlman correspondence, can also be expressed nicely when it comes to computing correlation functions of the field. So I tried to explain that. 307 00:47:06.830 --> 00:47:16.319 Courant Events Right: And, to do that, it's very nice to repackage the information contained in this thermunic model using a different 308 00:47:16.320 --> 00:47:29.259 Courant Events Right: kind of a different bond improvisation machine, which is the following. So, we're going to introduce what's called the Majorana Fermion. Before, we were dealing with something called different Fermions. Personally, I knew nothing about this, and I still, 309 00:47:29.920 --> 00:47:31.980 Courant Events Right: I'm really not an expert in these things. 310 00:47:32.680 --> 00:47:45.960 Courant Events Right: So, what that means is, we take the derivation psi 1 and psi2, and we repackage them differently. So we're going to call psi the sum psi 1 plus i psi2 and psi star psi 1 minus i psi. 311 00:47:46.100 --> 00:47:53.649 Courant Events Right: And together, I think of this as another, you know, it's the same information, but repackaged different types as well. It's the same thing, I mean. 312 00:47:54.160 --> 00:48:02.959 Courant Events Right: And it turns out when you do that, one way of expressing the correlations of the sine golden, or the derivatives of the sine golden. 313 00:48:03.130 --> 00:48:08.760 Courant Events Right: is as expectations of this measure on the Ferm loans for the Fermian measures. 314 00:48:08.990 --> 00:48:26.759 Courant Events Right: So that means, if I look at the correlations of the holomorphic and other holomorphic derivatives of the Sang-Galdon models, they are given by the expectation under this hermeneic measure of the product of the psi log psi star, depending on whether Si is 0 or 1. 315 00:48:29.900 --> 00:48:30.950 Courant Events Right: Under this motion. 316 00:48:31.590 --> 00:48:34.399 Courant Events Right: And again, you have to understand this in the instance. 317 00:48:35.970 --> 00:48:40.449 Courant Events Right: Okay, and this, you know, bracket notation, this expectation under this measure, 318 00:48:41.530 --> 00:48:45.110 Courant Events Right: And it turns out, by the way, that this has very concrete expression. 319 00:48:45.370 --> 00:48:53.310 Courant Events Right: These correlations, so… Okay, so… Why on the right-hand side not bilinear the Fermions? Is that the… 320 00:48:54.880 --> 00:49:04.299 Courant Events Right: Why is the right-hand side not binary? Usually it's a side-side bar or something. Yes. Oh, I guess maybe it's… probably, probably, yeah, sorry. 321 00:49:07.580 --> 00:49:11.470 Courant Events Right: So… 322 00:49:11.960 --> 00:49:23.020 Courant Events Right: one can go further, because, you know, just like, for probabilists, you know that if you have a Gaussian number involved, you can compute 323 00:49:23.050 --> 00:49:42.269 Courant Events Right: the nth moment of aggression in terms of its two-point functions. So that's the weak rule, and it turns out there is a similar property for Fermionic fields, so I guess it's called the Fermionic weak rule, and as a consequence of Coleman's correspondence, it's possible to evaluate n-point function 324 00:49:42.390 --> 00:49:48.040 Courant Events Right: Well, of the derivatives of the same double field in terms, just of its two-point functions. 325 00:49:48.210 --> 00:50:06.180 Courant Events Right: And this gives exactly the following fact. So, yeah, endpoint correlation functions of, polymorphic and anti-polymorphic derivatives of the field is a determinant of its two-point functions. So the fact that it's a determinant express the fairness in nature of the 326 00:50:08.890 --> 00:50:27.419 Courant Events Right: So this hermic Greek rule is very convenient for the identification of the same garden, the same input, that we're considering for the Daimler model, because it means that, you know, it tells us that if you want to show something in the same garden field, you have to find determinants for the, 327 00:50:27.530 --> 00:50:33.250 Courant Events Right: Correlation functions of the derivatives of the hat function, and that's exactly what we want to know firstly. 328 00:50:36.010 --> 00:50:41.149 Courant Events Right: Okay, so this was kind of informal so far. Let me tell you what is rigorous. 329 00:50:41.460 --> 00:50:59.399 Courant Events Right: So, in the Unit disk, if I take omega equals, yeah, the Unity disk, and I fix some smooth function with compact support, the very recent work of, Barb, Hertanen, and Webb, following the earlier work of Baro Schmidt and Webb, the brain. 330 00:51:00.310 --> 00:51:07.509 Courant Events Right: construct the Sandgarden model at discrete Fermion point, and rigorously establish the Coleman transform. 331 00:51:09.740 --> 00:51:16.689 Courant Events Right: So, more precisely, they start from the wideness regularization of the gastric field, I call that piocider. 332 00:51:16.810 --> 00:51:21.670 Courant Events Right: And you can introduce a weak, renormalized cosine of the field. 333 00:51:21.970 --> 00:51:25.159 Courant Events Right: So this is what I indicate with these two dots. 334 00:51:26.240 --> 00:51:35.180 Courant Events Right: And, so you take the cosine of the, oops, I guess the square would be that is missing, from the right-hand side. 335 00:51:35.550 --> 00:51:51.660 Courant Events Right: So you would take the cosine of the regularized field, you normalize it appropriately, as I said, and, yes, that's the weaker normalized field. Now, it turns out, what, if you want to see, something further for the, 336 00:51:51.910 --> 00:52:03.040 Courant Events Right: or at the free founding point, you have to renormalize further, and that's this double dot that you see on this slide here. In the paper, there are two very big dots, I don't know how they do it. 337 00:52:03.230 --> 00:52:07.689 Courant Events Right: So that, with my software, I put on this, so sorry. 338 00:52:07.940 --> 00:52:18.159 Courant Events Right: So you have to subtract some, some additional constant, which, which is explicit and is universal, whereas, alpha, which really should be zero. 339 00:52:18.480 --> 00:52:22.769 Courant Events Right: But Sioux Mission, was explicit, but not present. 340 00:52:23.780 --> 00:52:36.940 Courant Events Right: And, right, so the, you know, right, the expectation of F for the renormalized, sign-old model at the renormalization scale of sine is given exactly by this, expression. 341 00:52:37.970 --> 00:52:48.339 Courant Events Right: Okay, so you have to, you have to subtract mutually, addition, and that's related to the fact that, the sand garden at speeda equals 4 pi is not, 342 00:52:48.540 --> 00:52:50.679 Courant Events Right: Absolutely continuous respect to the dashboard. 343 00:52:53.260 --> 00:53:02.650 Courant Events Right: And then the theorem of value is that as epinell n tends to zero, if you look at the derivative of this field, sorry, I will simplify. 344 00:53:02.860 --> 00:53:13.930 Courant Events Right: If you look at the correlations, of these… of the derivatives, you know, holymorphic and other are not equivalent to the field, then this, exists, and it's given by the terminant. 345 00:53:15.400 --> 00:53:26.040 Courant Events Right: And, the determinant of the two-point function. And this two-point function, you know, you may understand this quite well. It's, particularly it solves, 346 00:53:26.270 --> 00:53:30.319 Courant Events Right: A certain explicit boundary variable, certain explicit boundary permission. 347 00:53:30.560 --> 00:53:38.870 Courant Events Right: Say again? Yes, beta equals 4, 5, yeah, I should have said. This is beta equals 4-5, and yeah. 348 00:53:39.750 --> 00:53:51.699 Courant Events Right: So, so if you want, it constructs the Sandgarden model associated to this profile row in the UB disk with beta equals for pi, and shows the validity of the Kolman transform. 349 00:53:58.240 --> 00:53:59.850 Courant Events Right: So, how much time do I have? 350 00:54:01.170 --> 00:54:02.590 Courant Events Right: open traditionally… 351 00:54:02.700 --> 00:54:12.670 Courant Events Right: You're the chairman. A bit less than 10 minutes? Okay. I tried to tell you a little bit about, 352 00:54:15.170 --> 00:54:23.570 Courant Events Right: Yeah, I'll try to tell you a little bit about some of our methods and some, some, some things related that I think are related to this, 353 00:54:23.950 --> 00:54:28.890 Courant Events Right: sandboard model in general, and which is a massive homorphicity, so… 354 00:54:29.560 --> 00:54:43.200 Courant Events Right: The first thing that we have to do is, you know, up until now, I was kind of thinking of my vector field alpha, well, as a vector field, but now it turns out it starts to be better to think of it as a complex valid function. 355 00:54:43.690 --> 00:54:52.049 Courant Events Right: And, we defined massacromaticity as the equation that says D bar F is half alpha F bar. 356 00:54:52.700 --> 00:55:00.449 Courant Events Right: So, you know, if alpha equals 0, we find the equation d bar f equals 0, which is the equation for holomorphic. 357 00:55:02.500 --> 00:55:14.069 Courant Events Right: And this generalizes the notion which was introduced earlier by Michael and Spirnov, and had been studied also by Parton in the constant case, where alpha was a constant of purely imaginary imaginary construct. 358 00:55:16.780 --> 00:55:21.550 Courant Events Right: So here's a lemma which is, 359 00:55:21.880 --> 00:55:32.169 Courant Events Right: kind of, interesting, maybe, which, relates some of the notions I've introduced so far, which is that if I take an alpha-massive holistic function in this sense. 360 00:55:32.370 --> 00:55:36.380 Courant Events Right: Then this real part is massive harmonic. 361 00:55:36.500 --> 00:55:49.430 Courant Events Right: Meaning, you know, mass here with respect to the mass, which is exactly this function that I assume to be non-negative, for the potential. So, meaning that, includes the Laplacian 362 00:55:49.430 --> 00:55:57.139 Courant Events Right: of the real part of F, and I find that it's mu times, this real part, where mu is this, 363 00:55:57.320 --> 00:56:01.950 Courant Events Right: mass function, so there would be some… the Laplace energy. What's the current? 364 00:56:02.130 --> 00:56:05.920 Courant Events Right: Is this a Dirac equation here? I think so. 365 00:56:06.080 --> 00:56:10.050 Courant Events Right: the D bar F equals alpha n-bar. Yeah. 366 00:56:10.590 --> 00:56:17.099 Courant Events Right: I'm not sure, but for me, the Dirac equation would be, let's say, for a function, so… 367 00:56:20.590 --> 00:56:26.270 Courant Events Right: Could we that, when you apply, I don't know, I don't. So, basically, I don't 368 00:56:27.810 --> 00:56:33.810 Courant Events Right: So, if you take, so, as I said, so the dilemma is that if I take something which is alpha anomorphic. 369 00:56:33.980 --> 00:56:37.500 Courant Events Right: Alpha mass equals more than its real part is massive harmonic. 370 00:56:37.730 --> 00:56:44.370 Courant Events Right: I don't say anything about the imaginary part, because it plays a different role in the story. 371 00:56:45.030 --> 00:56:57.400 Courant Events Right: We do not know if, conversely, starting from a massive harmonic function, does there exist a function, you know, does it have a harmonic conjugation? 372 00:56:57.520 --> 00:56:59.969 Courant Events Right: Does there exist? Can I turn it? 373 00:57:00.110 --> 00:57:16.499 Courant Events Right: by adding an imaginary part, and I turn it into a massive polymorphic function. We don't know if that's true in general, but fortunately for us, it turns out that will be the case of what we really care about, which is an inverse custom matrix that I'm going to talk about now. 374 00:57:19.860 --> 00:57:37.509 Courant Events Right: Okay, so a bit of Castellan theory. So, as I said, you know, the Sngoordin model is not conformal, but it is integral, and for us, I think, you know, parts of the integrability comes from the fact that the dimer model itself is something that has a lot of structure. 375 00:57:37.510 --> 00:57:41.940 Courant Events Right: And this structure comes from casted into wood, so… 376 00:57:42.700 --> 00:57:58.489 Courant Events Right: Basically, by the definition of the right function, I know I didn't believe the definition of the right function, but you have to trust me on that, the discrete derivatives of the right function corresponds precisely to whether an edge is occupied by a diamond. 377 00:57:58.980 --> 00:58:06.999 Courant Events Right: And so, if you want to understand the correlation of the derivative of the right function, then you have to understand dimer-dimer correlations. 378 00:58:08.060 --> 00:58:16.650 Courant Events Right: Now, gasoline theory is exactly, what is telling you what, what, what you need, what, what you need in order to compute, 379 00:58:16.750 --> 00:58:32.760 Courant Events Right: And it's telling you that the probability that you see a particular set of edges appearing in your random matching is given… well, actually, there is a constant, like, expense to 1, not rightly. 380 00:58:32.910 --> 00:58:45.289 Courant Events Right: is given basically by a determinant of the inverse of this celebrated matrix, the Castellan matrix, which you can think of as being a signed adjacency matrix of a graph. 381 00:58:45.840 --> 00:58:47.209 Courant Events Right: whether it's inverse. 382 00:58:47.530 --> 00:58:48.850 Courant Events Right: First one, please. 383 00:58:50.460 --> 00:59:00.329 Courant Events Right: Okay, so the key is really to show the convergence of this inverse cascade matrix to a pair of complex functions. 384 00:59:00.330 --> 00:59:12.009 Courant Events Right: And, you know, the point is that if you take your light vertex B to be on the blue lattice, then you will get a certain limit, whereas if you take it to be in the red sub lattice. 385 00:59:12.010 --> 00:59:27.410 Courant Events Right: you will get a different limit. So, the limit of the inverse scale matrix kind of depends on the combinatorial type of the black vertex, and in my mind, this corresponds really to the fact that at any point in the continuum, you have two directions, upside 1 and upside. 386 00:59:27.840 --> 00:59:40.020 Courant Events Right: So, you know, this sort of a kind of funny combinatorial construction, where you impose this temporalian structure for the domain, which separates the two types of black vertices into two. 387 00:59:40.020 --> 00:59:48.479 Courant Events Right: you kind of see it reflected in the fact that, naturally, for the, Sang-Gordon model, there are two types of Fernions, the Psi 1 and Upsai 2. 388 00:59:51.410 --> 01:00:04.279 Courant Events Right: Right, so as I said, the key is to say something about the scaling limit for the numerous gas-dependent matrix, and what we show, really, is that, yes, I'm just basically saying what I've just said now. 389 01:00:04.480 --> 01:00:08.740 Courant Events Right: that there is a scaling limit for this inverse SMM matrix. 390 01:00:08.850 --> 01:00:16.400 Courant Events Right: And, that deal depends on whether you're talking about a black vertex of style blue, or of type red. 391 01:00:20.010 --> 01:00:36.259 Courant Events Right: So, let me tell you a little bit, a few things about how one pulls out. First of all, it's not hard to see that when you consider a black vertex, which is a blue vertex, then K inverse is a discrete derivative thing of the massive brain function. 392 01:00:37.180 --> 01:00:45.519 Courant Events Right: So, using tools from discrete complex analysis, which are extended to the massive setting, this gives a scaling limit 4K inverse of this situation. 393 01:00:45.820 --> 01:00:52.420 Courant Events Right: So, it's more complicated to analyze the case where the black vertex is basically a red vertex. 394 01:00:52.730 --> 01:01:04.019 Courant Events Right: So, when alpha equals to 0, you say KK inverse is the identity, and basically one way to interpret this is to say that k inverse is a discrete holomorphic function. 395 01:01:04.360 --> 01:01:13.110 Courant Events Right: So, you know, you kind of read this identity to say, well, the derivatives in Y are exactly i times the derivative in the X direction. 396 01:01:14.240 --> 01:01:26.680 Courant Events Right: And so, when you integrate, these derivatives together with the earlier asymptotics for the, you know, the case where X was a blue vertex, this, you know, allows you to, 397 01:01:26.830 --> 01:01:32.119 Courant Events Right: derive a conclusion that the inverse casting matrix has a scaling limit. 398 01:01:32.270 --> 01:01:35.890 Courant Events Right: You know, whether or not you are considering a mover at the accelerator. 399 01:01:37.420 --> 01:01:50.290 Courant Events Right: Now, in the massive setting, this is not so simple, unfortunately. So basically, this was… this argument here that I just sketched was essentially Rick Kenyon's original argument for his proof of conforming bindings. 400 01:01:51.090 --> 01:01:52.100 Courant Events Right: Doug. 401 01:01:52.400 --> 01:01:56.380 Courant Events Right: Now, in this massive setting, this is not so simple, unfortunately. 402 01:01:56.840 --> 01:02:10.170 Courant Events Right: But basically, in the continuum, what you can say is that if you have a function which is U plus IV, and it's massive polymorphic, then you can rephrase that in terms of, basically a version of the Cauch-Hilman equation. 403 01:02:10.760 --> 01:02:20.509 Courant Events Right: which relates, the gradients of N and the gradients of V, by rotation, and that, you know, you have to conjugate by, 404 01:02:20.800 --> 01:02:21.779 Courant Events Right: with potential. 405 01:02:22.860 --> 01:02:33.209 Courant Events Right: So that's in the continuum, but what is key to our work is that we find an exact discrete form of this identity, which is satisfied by, came close. 406 01:02:33.370 --> 01:02:50.020 Courant Events Right: And okay, so that means that, essentially, once you understand its identity at the discrete level, you can use asymptotic for K inverse, restrict it to blue vertices, and transform it to asymptotic for K inverse on red vertices. 407 01:02:52.330 --> 01:03:09.490 Courant Events Right: All right, so thanks, thank you here, and I'm just going to advertise that there's a program at the Schrodinger Institute. This is more about connections between probability and combinatorics, but if you're interested, then please let me know. Once again, thanks for your knowledge. 408 01:03:15.710 --> 01:03:17.730 Courant Events Right: Questions? 409 01:03:20.640 --> 01:03:22.279 Courant Events Right: But you do, Mike. 410 01:03:24.160 --> 01:03:41.150 Courant Events Right: Yes, it turns out, thank you. So, it turns out we can't really solve the moment problem, meaning that these moments really do determine the distribution uniquely, and we find the… 411 01:03:41.390 --> 01:03:43.009 Courant Events Right: An interesting formula. 412 01:03:44.580 --> 01:03:47.460 Courant Events Right: There's an interesting, formula, where is this? 413 01:03:47.790 --> 01:03:54.139 Courant Events Right: I'm not gonna move for it. There's an interesting Freedom regularized determinant formula for the… 414 01:03:54.310 --> 01:04:02.509 Courant Events Right: For the moments, basically. So, that says that the distribution really is determined by the moments, because in fact, it's determined by its cumulance, that's what I should say. 415 01:04:03.800 --> 01:04:06.180 Courant Events Right: But the moments determine the killings, so… 416 01:04:08.990 --> 01:04:11.800 Courant Events Right: Let's just make appointment. You got it. 417 01:04:12.150 --> 01:04:26.410 Courant Events Right: away from that… Yes. Yes. Right, right. 418 01:04:26.730 --> 01:04:37.760 Courant Events Right: Just to make sure that around for 5 minutes. 419 01:04:38.530 --> 01:04:44.070 Courant Events Right: Right, right. Exactly, that's it. 420 01:04:46.170 --> 01:04:47.270 Courant Events Right: You don't appreciate it. 421 01:04:51.420 --> 01:04:55.489 Courant Events Right: The basic model is very practical, but also way too difficult. 422 01:04:55.820 --> 01:05:02.410 Courant Events Right: Yeah, it's German. 423 01:05:02.510 --> 01:05:07.110 Courant Events Right: That's a good question, Alan. 424 01:05:07.380 --> 01:05:10.720 Courant Events Right: I think strong relations to business school. 425 01:05:11.560 --> 01:05:13.949 Courant Events Right: Yeah, that's, like, a very interesting program. 426 01:05:15.070 --> 01:05:25.709 Courant Events Right: And I should say that, you know, the paper of, part of, they really specifically relate, the same garden model to the limits of, 427 01:05:26.590 --> 01:05:32.650 Courant Events Right: temperature that you have this model. Probably is an expression of this, that modernization. 428 01:05:32.840 --> 01:05:45.000 Courant Events Right: But as I said, you know, the design customization, that's the one that I know of, which comes from this paper of Jumeda that was alluded into earlier, behaves very highly as a primary. So I don't know… 429 01:05:45.100 --> 01:05:55.820 Courant Events Right: to what extent, you know, certainly feels right. Technically, I'm not sure, and I haven't thought that all about, trying to communicate things that that's what I'm gonna do. 430 01:05:57.220 --> 01:06:00.919 Courant Events Right: It's not the same as he's at. It's starting to be currently on the way from… 431 01:06:01.180 --> 01:06:07.719 Courant Events Right: I mean, it's like a 6 vertex one with the mask. You should, you should speak to them. 432 01:06:08.910 --> 01:06:10.080 Courant Events Right: Thank you. 433 01:06:12.860 --> 01:06:23.709 Courant Events Right: There is one last question. Yes, I'm very sorry if I may come in. So, in fact, in this paper by Julianne, he is, like, 15 years old, there is a response which she says, I'm the doctor. 434 01:06:23.810 --> 01:06:36.320 Courant Events Right: This is the American Greek class of the race, which I know they're… Exactly, it's not entirely. And, all the graduation identities, they call them. 435 01:06:36.320 --> 01:06:46.539 Courant Events Right: Yes, but I think from the downer side, it's a kind of boundary conditions that you end up with, they are very minimalized. 436 01:06:46.770 --> 01:06:49.370 Courant Events Right: Nowadays, end up eating well verifications. 437 01:06:56.380 --> 01:06:59.169 Courant Events Right: Great, so thank you very much for your touch. 438 01:07:08.540 --> 01:07:10.879 Courant Events Right: Rob didn't pop for me.