 thanks for the introduction and also thanks to both previous speakers for actually introducing many notions I needed my talk. I should say that I find myself in kind of a difficult position because it was more an introductory talk by Ugo and then really advanced by Dini. So maybe the kind of viewpoint you could adapt on this talk is just to understand in the time in a reasonable detail why the life is so nice for the ISIN model and how the structures which Dini was talking about how they can be used to prove probabilistic theorems something like that. Okay so I should say that this part everything but the very last line is something okay I and my colleagues we told about many times okay we'll see how it goes here is maybe something something new time permits and I'm going to to say something new. Okay so just to have a picture in mind first please change the notation back I apologize for that I'm working with the ISIN model and spins are now again live on faces yeah that's sorry for that that's not very classical your shame on me but okay spins on faces and it's going to be always behind is going to be a question of universality. So okay for a while you might think about the ISIN model on Z2 so basically I take a planar graph a finite planar graph which I'm going to not take omega delta okay I took it uniformly at random it happened to be a piece of Z2 so just think about Z2 I'm going to think about the critical model in the most part of the talk so in the on the square let's say the critical value okay so let me first define the model so okay we saw it already several times what is that this random assignment of pluses and minuses yet again to faces of my graph so basically here spins live and the probability to see concrete configuration is proportional to exponential what I never know the sign to beat some overall okay I need some notation for faces typically faces are going to be denoted by letter u so u and u plane and here it's going to be some interaction constant think of it as a given positive number say for a homogeneous model okay all this all this interaction constants they may might be taken is one okay some interaction constant and then what okay this is a random assignment of of spins to to the faces of my finite graph and I am going for you know go assume that everything at the boundaries plus work with plus boundary conditions this can be relaxed for instance you can play with combinations of plus and free boundary arcs okay but let it be plus also well I need some other notation you can rewrite it like that and this is pretty much pretty much actually what we go already did so you can say this is some other normalizing constant but then to compute the relative probability to see this particular configuration you simply multiply overall edges you multiply either once or some non-trivial parameter x okay effectively x is simply exponential minus 2 beta okay this is my parameterization and this is the link with the o n model you go basically already explained this morning okay so now coming back to the criticality it was mentioned that on the hexagonal grid the critical value of the parameter is 1 over square root over 3 on the square grid okay criticality on z2 homogeneous model x is 1 over square root plus 1 square root 2 plus 1 on the hexagonal grid faces of the hexagonal grid x is 1 over okay by the way this is tangent of by over 8 and this is tangent but you can also have in mind some more general picture so okay first of course it could be triangular lattice then if you know what rhombic lattices and isogradial graphs is the invariant model on isogradial graphs is okay just this picture but actually one of the goals is to understand really the universal behavior of this model so you might think of the following of the following setup you just say start I'm going to erase it and don't comment much on that you start with z2 but instead of considering the homogeneous model when all the interaction constants are the same you fix some fundamental domain put here x1 x2 x3 x4 5 6 7 or maybe even 8 9 10 and then multiply it periodically in all in both directions then it's known what is the criticality condition again due to Hugo and David Simazzoni and Zongian Lioz in this setup so you might wonder okay how to see say conformal invariance in this setup then when you start thinking you immediately get the following question okay but if they're unbalanced then there is no hope to see conformal invariance if the lattice is drawn this way so it might be for instance that horizontal and vertical direction are not equivalent right it should be some tilt if you think a bit more you understand that it kind of reduced just to tilt this way so you really should find a way to embed your fundamental domain into a plane I don't know like a parallel diagram and then do something like that again you might wonder what would be a good embedding etc so time permits I want to comment a bit on that this is exactly the subject of the very last item so okay for a while it was considered as an open question and now there is some hope okay but to start with just again some abstract graph I don't even assume at the beginning that the interaction constants are critical so okay just some combinatorial description so this combinatoric spot what is the basic object in the isin model is the order disorder correlators discussed by Dini so in this particular case what does it mean so I'm going to work with such objects so my spins they live on faces my disorders are going to be are going to live on vertices like I have a product of say m disorder seven spins and I'm going to not a the correlation function like that the very first remark is that it can be solved probabilistically so yet again what's that given say five oh no five sorry four disorders like v1 v2 v3 maybe before here I can fix disorder lines they were originally blue in the need talk in stock but okay then become orange something like that so I fix these order lines and then okay I just my spins they live here okay I can consider such a correlation whereby this product I can simply mean okay I can write it let me do it like that I can write this is an expectation of some random variable just to emphasize that there is a direct probabilistic meaning where this mu gamma is what gamma is a collection of all the spots I can choose whatever I wish and then this is exponential minus 2 beta I hope plus 2 beta j u u 3 so it means that along oh la la it was a mistake at the very beginning so there is no two yeah sorry for that oh la la la it was mistake at the very top right so it's now inconsistent so in the first line the alliance spins they contribute they must contribute exponential plus beta and the disaligned exponential minus beta and it turns out to be either one exponential minus two beta here okay and now what I'm saying what I'm saying with this with these disorders is that basically I flip the sign of the interaction constant along gammas okay it should be minus so it means basically that I replace plus beta there by minus beta okay so this is a probabilistic object and this story we use we use invariance under other topological moves of those strings as they describe basically it says that this does not depend on gamma chose until they do not cross spins okay the gamma is if is yeah is a fixed is a fixed path think about that between these four points here it must be even okay or you go to infinity or to the boundary in my case okay now what I suggest is a moving variant description actually you can view it as follows so that's why I prefer this notation so okay we know that there is a probabilistic meaning but now I want to think about that as a function of all the variables of all the phases and vertices and then you should be careful with signs because remember there was a sign change when one of the lines cross the spins jumps across the spin so because of that I prefer this is defined up to a sign but then you can think what does it mean in terms of double covers so what I want to say is that this is a spinner on a double cover of what I take all phases of my graph all vertices sorry of my graph power m all phases of my graph power m and then when I say spinner okay by definition in my language this is just a function which is defined on this double cover and which flips the sign between the two sheets of this double cover okay and the rule is that it has the same structure as the product of all u minus 1 to n m q 1 to n v p minus q one half this is just a function on the complex plane or in your domain which is defined up to a sign but you can think of it on the random surface so each time one of these goes around one of us you change the sign so this is the basic object then what you can do starting from here exactly as you did so you define when did I start by the way just to control time it was 22 35 okay thanks two plus three right so okay now what you can do is to define fermions actually so what are fermions okay first I'm going to use the notation chi just to emphasize that those are real valued I mean yet again okay that that's really a kind of down to earth version of the previous talk so just we view it like that okay what is what here so first what is C? C is what we're used to call a corner of our graph so C actually lives here okay so you take on this picture you take all the corners of all the faces of your graph okay so this is just a notation I'm going to use so it means that each time I write a chi C it means that I write both disorder and spin correlators okay with that you should be careful because effectively there is this branching structure so there is a choice of those four four corners around the vertex okay there is a branching structure just to comment on that if I want to consider a correlation function so kind of remark it's not going to be in many theorems maybe even no no theorems or that's like in the previous talk like remarks and proposition and properties okay remark is that okay if I think about this I'm going to introduce the notation along the way so if you are lost with the notation just cry and okay I'll come in so what's that I have one fermion like mu sigma and then I have a number of disorders and spins sitting somewhere so this omega is effectively a set of other vertices and faces so other vertices and faces right so I have such a correlation function and then I should say okay where it leaves okay is this is a spinner on something which I'm going to denote in an ugly manner okay let me comment on that so first what is a greck a greck is by definition the set of all corners okay so the set of all corners all corners and I'm going to be just combinatorial just abstract enough so it is square grid which is drawn but think of an abstract graph okay this is a set of all corners of g then what this cross means okay what first what this cross means this cross means that actually instead of having just this graph just this structure along every and around every edge what you have is a double cover it branches like that so it branches around an edge why because if you keep track of this v minus u one half it makes the full turn when you go around okay and equivalently it branches around every vertex and around every face so it is a double cover which is ugly it branches everywhere okay so this means branches everywhere and this means okay if it was a spin another spin inside here then it kills the spin participating in chi and the result is that or say disorder at v and the result is that the correlation does not branch around this point except over okay omega so technically this is varsigma okay yet again in absence of spins and disorders it branches everywhere because when you go around the vertex again this square root adds you an additional sign I still don't get what what you mean by double cover so it means that over each of the corners there are two points and then I ask how they are connected with each other that's like a combinatorial double cover and to specify that it's enough to specify either I have two two closed contours so I have a single contour here there and over all the faces and vertices and what I am saying is that this cross means that I have I have branching everywhere and this omega indicates that except at marked points is it better now okay good thanks for asking because that's somehow an important definition but how did your two sheets in your cover are glued together this is an abstract stuff so that's like an abstract theorem so that's enough to specify the monodramy around every point every face of the blue graph either this is plus one or minus one so combinatorially this is it's enough data so you you simply take you simply control I mean on which sheet you leave when you when you propagate okay so and now there is something important which I want to to write there just to keep later so there is a linear equation of motion the propagation equation for fermions etc there are many many names for that so just let me prefer to use the name propagation equation so what's that imagine we have just a local portion of our graph simply effectively just a single edge and it's a notation so what's going to be notation this is going to be say v0 v1 this is u0 u1 and also I have some corners nearby so the corners are going to be notated like that so the first index is the index of v and the second index is the index of u okay so and remember yeah that's a slight mess because they leave not on the graph g but on the double curve so then what I pretend is the following it was known at least back to 70s but effectively that's I believe already nonzaggers work so this I cannot claim okay then there is an identity so if I consider the function let me denote this function by x capital just to reflect that this is a correlation function with single chi so if I and let me keep other points in notation so if I consider such a function then okay it satisfies the following the following identity uh if I evaluate it say here at c0 0 then this is a linear combination of these two values namely again I'm introducing some notation along the way be careful so remember I had this parameter x somewhere there originally it was introduced in terms of the coupling constants but I can also use whatever other parameterization I find convenient right so and for this purpose I find convenient this parameterization okay so x e is tangent 1 half theta e that's also why I wrote there pi over 8 pi over s 6 okay you see eventually it's coming it's going to come into play with geometry okay so there is this identity what is the reason for that I'm not going to prove it of course but that's very simple so let me just indicate so basically you say okay let me consider the product of two disorders sitting here and there then I explained that this is a random variable so to evaluate it I can fix whatever cut linking v0 and v1 I can do this cut and then there is a way to write it in terms of spins and in terms of spins it's going to be x to the product of two spins and x is exactly that then okay the product of two spins there's a specific feature of the ison model it has only two values because of that there's a linear combination of okay of constant and sigma sigma and namely this linear combination is equal to I never remember of course 1 over sine theta 1 minus sigma u sigma u1 cosine so that's a trivial computation okay I changed the parameterization but at least it's clear that some linear equation must be there and then to get so what I now have is a linear equation for the product of two disorders for the constant and sigma sigma and what you do you multiply it by by sigma u0 sigma v0 so you now multiply it by sigma v0 and then okay what you see eventually is that chi 0 0 appears there and okay this this equation must appear okay so a kind of very simple thing but that's a very restrictive property of the ison model effectively it means that if you know like the values of those function x along the line then effectively you know all the values because once you know this and that you immediately know that and then you can propagate further along along the edges okay that's why propagation equation now okay the question how homomorphicity appears maybe I still I still write it here so how to pass the homomorphicity at this point it must be some at least it could be not must but it could be a kind of inconvenience with this weird branching structure so you might want to kill it right and to work with something with something easier so the way to kill it and let me that's important let me notate it somehow extra to kill it you do the following so you say all right let me introduce the following object so to evaluate it depends on on the corner it leaves again on a double cover and to evaluate it on the corner you say effectively I take the square root of v minus u back so I prefer to have it unimodular so that's effectively the square root one of the square root of v minus u but that's a unimodular number and for some reason let me put here a constant factor don't ask it don't tell I mean okay just think it's one it works until some point very smoothly and then this is just a change of the notation okay this is something which again branches here everywhere so because of that you what you can do you can introduce the function f which is by definition just a product and it's nice because these branchings they kill each other right now when you go around around whatever an edge or around the vertex both factors branch and because of that the product is well defined so now this is a spinner in crack omega all over okay let's let's leave it how it is on that and what is that this is now branches only over omega okay I hope the notation is okay not weird enough that that's a question how to develop it okay and now you can ask all right now I have some observable by the way it's now complex valued now you can ask what are the linear relations for this observable and then the answer is pretty nice provided there is a link between weights and the geometry of your embed so right now okay assuming let me let me write it here so now I assume that our graph for instance it was indeed the square lattice with the critical parameter so I assume that or a hexagonal one that locally we have a rhombus so that this is a rhombus and it happened by a miracle that this angle is a is exactly the same theta so that this theta is exactly the same then what you are seeing is that okay not not immediately but you can work it out then this condition can be written as follows so now I have a function which leaves here etc okay this can be rewritten as the discrete or homophistic condition you can always consider such an embed not at all no no no no that's a that's a very good question so it is not clear and this is totally wrong I mean so right now this is a very very special case so this is very very special assumption but at least on regular lattices we are there and what we see is that okay some discrete homophicity pops up so what I want to emphasize is that this way you see it not from combinatorics but just from that propagation equation so I didn't didn't draw any contour expansion so you can draw contour expansions of the correlation functions and deduce it from there but effectively this is okay just a simple consequence so dolly this is more weight I mean I still have 30 minutes I should adjust you know the the plan because otherwise we would have up exactly in the same manner I don't know it's going to happen that's the subject of the very last point in the discussion okay so we have some discrete homophicity then what you can say is that all right but here there is an information so remember you go lecture so there is a hidden information this f is not an abstract complex number right because this guy is real it is something fixed so effectively there's a complex number with a prescribed complex phase and in this equation it is not it is not remembered yet but then okay this was I don't know like lemma and then okay that that's the first the first claim and the second claim is that you can rewrite it so there exists a number which I would denote fz so what is z z from now on is simply a quad you might think about its center if it was a rhombus then this is a center so such that okay fc reads as effectively oh la la sorry sorry that started wrong so this was this was real and okay like that so this quantity has a complex phase eta right if it was as I tried like eta bar plus eta then it would be real and right now this is parallel to okay so effectively those there's four numbers they can be viewed as projections of a single number fz and moreover there is a kind of a nice interpretation so so now I have a function which is which leaves on quads fz and you have a nice interpretation this is the following imagine I want to integrate that not along the the whole boundary of the rhombus but just to compute two increments here and there okay so I compute c 0 q u q minus e 0 plus f 1 q v 1 minus u q they're the same for both choices of q because I can integrate along any contour I can move the contour and if I multiply it by two then okay what I see this is a computation is fz v 1 minus v 0 just an exercise so what does it mean it means that effectively instead of considering integrals of fc along the boundaries of rhombus I could also consider the contour integrals of my fz like that they all also vanish and the same holds if I would replace vertices by faces because somehow that's okay that's a dual picture so similarly for you and similarly for you okay and similarly for fz and this gives you a tool because now okay you have a function with okay all the possible contour integrals vanishing and then you might hope to identify the limit of these functions when the mesh becomes small so now just to indicate what is the setup for convergence theorems so what you now can do is default so what is the goal of I mean one of the possible goals in this activity so the goal is convergence theorems okay first it must be some parameter the parameter is the latest step it must be some setup which is going to be what we call scaling limit this means we have an approximation to a given domain say by square grids and the criticalizing model there and what we want to know is the following statement the expectation of the product of some variables okay what are the variables until now those are spins disorders those fermions okay let me maybe instead of expectations I just put this notation like I have psi psi bar sigma so okay what are what is psi psi is a linear combination which corresponds to the function f so this is going to be psi z this is going to be psi z bar okay so in discrete all these quantities they can be viewed probabilistically so if you're okay you might think just about single spins or just I don't know some mixture of spins and disorders and you want to prove that being normalized properly to some exponent these correlation functions they do converge to to some other some functions so those are just functions in omega which satisfy some okay nice properties what basically is this list of properties uh yeah I'm not going to write them all down but just to indicate so for instance each time you see this one it must be holomorphic in z and this is reminiscent of the fact that f is holomorphic this one must be anti-holomorphic then effectively you want fusion rules so you want to specify what happens when two points inside of these correlation functions approach each other okay number of fusion rules effectively you want that mu and sigma being collapsed they give you a fermion back so I'm just you know a bit hand waving uh here ask too much too many questions please so what's that this is omega psi psi plus psi bar then you want that when you collapse the fermion onto a spin the disorder appears because that's what happens in discrete right if I consider a spin multiplied spin here multiplied by the fermion here okay spins cancel out only only disorder survives so that's mu with some complex factor which I don't remember and so on and so forth and so on and so forth so okay you open the text book in conformal field here it is the basic example and then you learn what is the axiomatic this functions must satisfy what is their main property of those is that they are all conformally covariant so okay they satisfy conformal covariance property which means that when you change the domain all these functions they transform nicely they are multiplied by some powers of derivative of the conformal map so if you apply a conformal map to here so if you start with with omega and map it to some omega prime then effectively when you have when you have like some field sitting in z one etc some other field sitting at z n compute this in omega this is the same as you would compute in the new domain in omega prime up to a product of of the derivatives at z k to some power k and derivatives bar to some other so this is one of the possible goals just to say that the picture we have in discrete so somehow what is on the left you can view it as a particular case of the construction would go described in the morning in discrete so it defines these objects in discrete this is a picture you can learn from theoretical physics that it must be a set of correlation functions and what you want to do is a mathematical you want to prove that okay in fact discrete objects they approximate continuous ones okay so right now this is a theorem in the generality written here and moreover you can also add energies etc so that's really fixed you can play with boundary conditions instead of plus ones you can take a combination of plus and three you can do it in multiply connected domain so we stopped short before considering the rim and surfaces but they're okay you you additionally sum overall spin structures and this is I should say that this result due to Clemence-Anglaire, Konstantin Yurev and myself of course this is a very sketchy description because I didn't say what are deltas right and there were even two instances of deltas it was delta scaling exponent there and then delta plus delta minus scaling exponents there okay just to maybe to complete this picture repeat what are the exponents so for the spin and disorders this is 1, 16, 1, 16 and for the fermion for the psi this is one half one half zero for psi bar this is zero one half okay and at that point here I'm not going to really explain how to prove this theorem but the message you might take home is that okay somehow this description is completed so you really see the correspondence maybe it's worse to comment on a single point so I said that there is a main tool the main tool are those functions f okay you see some discrete homomorphicity and then according to the program written at the top so you should consider some boundary value problems for them okay so the the idea is to so the main idea is to exploit boundary value problems for these observables I don't have to comment on how working with fermions you get spins at the end this is an interesting question but here probably I don't have time for that instead I want to just to because there are questions about these S embeddings and people you know an organ the organizer wants to learn something new so I want to comment on a tool which was considered technical but in fact now I believe it's not so there is some construction which is due to Smirnov so in a sense what I explained till this point I mean the construction of these functions well it was there for years okay it was more a question how to use it and then it was a tool just the combinatorial stuff okay just a tool introduced by stars which is the following observation just imagine I start with whatever solution to this equation then you can do the following so you can define the function on both vertices and faces traditionally it's called h okay this is either hx or I'm going to meditate or hf okay think of some function which satisfies this this relation basically this is a correlation function and then you do the following so you say that given a vertex and a face so given a picture like that so this is v this is u this is c you simply say that h v minus h u is what in that setup it's going to be real valued and it's going to be x squared so here is a definition okay this is an exercise to check that it is well defined but this is not a consequence of the rhombus stuff it is well defined just because of this definition so what is important I want to emphasize that it is an abstract relation so to define such a function I do not need to fix whatever in Bayesian so what I need is well I just consider this is a map this fixed up to homotopy okay okay and moreover there is an analog of this statement so moreover okay you can say that if okay moreover well let me put it like here moreover in the rhombic case so this is abstract but in the rhombic case you can also say that uh h v1 minus hv0 maybe with the coefficient 2 again is imaginary part of fz squared v1 minus v0 so this function h it has some interpretation in terms of the function f and this is the moral of the story so okay now how the proof goes just in in two minutes okay you have a function which is holomorphic inside accepted singularities okay assume you know it is regular enough this is a kind of a tricky point because time to time you do not have any probabilistic argument like russ-semer-wilch theory for that so you did use it from just purely from complex analysis but assuming it is regular enough by my error theorem as we go explained it must have some limit properly rescaled maybe and then the question is what are boundary what are the boundary conditions and for the boundary conditions okay maybe in this color so let me put it here so what are boundary conditions for the function h those are just Dirichlet ones nerking by the way if you I mean if you combine this and that you see that boundary conditions for f for the function f are really ugly to work so okay h apparently happens to be a good object so yet again a priority when you define it like that it doesn't to be very much linked with resizing model correlations it seems to be just a technical tool but at least what is one good role is that it encodes boundary conditions in a in a nice manner and then okay you do some analysis here I don't I don't want to comment on this uh today uh here after all this theorem is at least for spin correlations this is five years old and even more so okay so but yeah right now here is a set up and what I want to do now is to jump to the last point just I'm not going to explain how to pass from fermions to spins and just go to the last point and to discuss a bit university so what is on the blackboard on the blackboard if you trust me that this program can be performed can be realized then there is a kind of a universal statement because it looks like everything works at least simultaneously on the square grid and on the hexagonal one right because what I only need is this condition which is purely geometrical okay by the way so if it was a square grid then theta is pi over four and then this x is exactly tangent pi over eight so it fits if it was hexagonal grid then this theta is pi over I don't know three and that is pi over six so it fits also if you know what is our radiographs are then okay this is z invariant model on on the netherite radiograph critical one and the proofs work more or less smoothly though just for the record I should say that for spin correlations still okay some ingredient is not written down and I cannot claim I really know how to fix it but at least for fermions it works well for curves before it works well so okay this universe but then there's a question okay Nikolai already asked and the question I already mentioned at the very beginning okay is it clear that given a graph I can embed it in such a nice manner okay so if you are lost you you might try to wake up because I'm going to to come back to the beginning of the story so somehow okay you count parameters and you see that there is no hope not at all and somehow you are disappointed because even in this doubly periodic case okay there is something puzzling I mean it should be possible to treat it but okay you don't have it in such a tool the problem is that this lemma seems to be very much dependent on geometry so now I'm going to comment on the way I hope all that can be relaxed so this embedding stuff and as a model example think about doubly periodic case what are s embeddings one of the statements in this criticality condition which I mentioned due to Ugo and David Simazoni is the following so in the doubly periodic so kind of a motivation the doubly periodic setup there are two functions two linear independent functions x1 x2 x2 x3 which are periodic and satisfying this propagation equation this is not how it's written in dear paper but it's equivalent way of saying so you just think about this propagation equation you want to find the periodic solution and then it happens that the kernel is not trivial effectively this is exactly the criticality condition so in the paper this is the kernel of the cuts work matrix but they are equivalent that's the same okay now there is the following idea which I believe to be new more or less partially okay some particular case was also discovered by March and Lee's independently in December okay now you can do the following you say okay I have such a function and it's still written I hope on the top that there is a way to define this function h just out of of this condition okay that's algebra so in particular it would work even x was complex why not okay you say now given okay given a graph a weighted graph and the pair x1 x2 spinners of real valued spinners satisfying this condition those are abstract ones they are not necessarily correlations of so satisfying you can try just to construct an embedding of your graph by using this function h this seems to be a bit ridiculous first okay just define s to be h x1 plus okay a bit weird and assume that the combinatorics of this embedding that there are no overlaps etc so I'm not going to discuss under which conditions this whole then I don't have a definitive answer but at least in some vicinity of is a radial embeddings this holds okay so it's going to be it's going to assign some value to whatever face to whatever vertex and face so it's going to be a picture like that and then you might wonder so the idea is that you want to generalize is a radial embeddings so in the is a radial embeddings all the squads they are wrong by okay then you ask but what is the properties of this quad if actually this should be a matter of computation right because you have functions probably tries by by two unknowns and the answer is the following so all these quads have happened to be tangential quads so the raw circles are inscribed into them okay then just trust me so claim all quads are tangential and moreover if you want to recover the weight this x e from the geometry there is a formula so there is something explicit okay in terms of geometry so there is a thing I want to insist on and the thing is that this correspondence is one-to-one so other way around if I draw you a tiling of a plane by these quads then it defines me a graph of course because I just okay up to a homotopy a collection of weights because I said that okay there is an explicit formula and moreover it defines me x x plus i x2 because they are simply square roots of those of those directions and they have correct branching structure so this is really one-to-one and somehow the ultimate goal of that is to use it to study is in model on whatever on more complicated graphs so ultimately on random maps okay I have no no clue how to do that but you can say okay I have a random map I equipped with the is in model and then I want to to to figure out what is a proper embedding so this is a kind of a candidate now of course the question is to which extent the relations above they hold right and the answer you said that not all graphs admit such an embedding right no I said that all admits all a priori okay that's a question what you ask about x1 and x2 so ideally you want them to be bounded right so that your tiling has so remember x is a square of the of the length so that your tiling is okay this should appear only at criticality so now it's a bit discussion part already so a priori you you cannot expect this this outside of criticality but on combinatorial level okay it works so now of course the the question is to which extent what is written on the left hand on the right can be generalized so and now let me tell you about this generalization to conclude so first okay let me add here an additional normalization so what is delta c so here is a vertex and a face and by this delta c I just define the length of this segment okay in the wrong by case it was everywhere the same now it okay now I'm going to correct just that equation to say what is true and what is not okay so this is a correction maybe here what is a good color okay yellow is okay blue here is a good color so this still holds true so whatever graph you start with provided you embed it in such a way apparently you see this critical morphicity appearing at least in some form so that's okay not very expected because okay at this point it's even a bit fishy because okay maybe there are several ways to embed the same graph and they are not all the conformal in equilibrium equivalent for instance you can put here instead of i2i it would be different embedding okay how to how to figure out what's happening then there is a correction to what to to this equation so remember I told you that the contour integrals of f of z are effectively the same as contour integrals around from by so in this case this is not not exactly the case there is an additional term okay so what is this function l so now it is not around us anymore and the function l effectively it measures the distance so lv minus lu is by definition simply distance v minus u okay in the rhombus case this disappears in the rhombus case this disappears it's simply zero and here a correction I mean you see correction and now you might wonder alright but maybe this fz is a better so by the way this still holds I mean this both still holds so still there is a number which is projected onto this fc and then you might say but okay what about contour integrals of that and the last piece of information I want to discuss is the following so just think about that let me draw a contour say on vertices of my graph something like that and let me try to integrate fz dz along this contour this is a closed form so effectively this is the same as the contour integral fz bar l minus l right and I can do integration by parts so I can always write it like fz1 minus fz2 bar multiplied by l really this is an identity so what I need is that this can there's integrals they approximately vanish what about function l maybe you expect that you can do you can make it really small so yet again what l counts it just counts okay when you go from white to black then it adds the length of the edge so imagine okay wishful thinking but this is a statement a fear statement saying the w periodic case that you can choose x1 and x2 such that this function is of order delta okay you have a hundred types of tiles so what you need is that in the w periodic case what you need is that when you compute the increment of l along this and along that they both vanish so what you want is that l periodic for instance in the w periodic case if this is periodic then this is of order delta okay but then what you immediately see is that you need a rather mild condition on f so what you need is that this function f effectively okay it's equicontinuous equicontinuity here is enough so once you know this tends to zero you know that all the contour integrals they vanish and okay you can you can play your game so here is the point that okay there is such a structure there are no theorems yet but somehow it looks rather promising uh yeah and just to comment a very particular case you can think about is the case discovered by March and Lee's independently as I mentioned when all the guys are deltoids so what are deltoids when all of them okay this is equal to that so it means that you have a circle pattern like that with different circles so in this case on one of the type of vertices the function l is simply constant so on the black vertices here l is constant so because of that you have half homophicity for free but that's somehow a particular case of the general stuff so now the current state of there basically is that okay it seems now to be a question of discrete complex analysis just to prove that f is regular enough and then it would open a way it would pave the way to really universality convergence here okay yeah I believe it's time to stop sorry for the delay are there questions so before that for iso radiographs yeah so we needed the control that the angles are not too acute that the angles are bounded below and there you believe that you can release the constraint to as you said no well that that that's a very good question so Nikolai I don't know at this point so somehow you know it was it was really open for years I mean how to handle how to handle something more general than iso radiograph stuff so okay so what marching proved for instance is that once you start okay it's written in less generality but the proof works in this generality if you start with whatever tiling such that the sizes are bounded the sizes of tiles are bounded time-to-time from one side but let us think from both sides then the model is critical in the sense of spin-spin correlations and the single point magnetization it's a good question to study now how much one can relax the assumption on boundedness for example if you take two nearby tiles if they are comparable this no comparable is too weak right because you still can have exponential growth no no no no what what you at least I mean intuitively what you should have is that the size of a tile a distance whatever r is all small of r at least otherwise this this doesn't look so okay for me that's more like you know a set of questions now because there is a tool so for those who know about this sub-super harmonicity of the function H there is an analog of that in this setup so there are a kind of replacements of Laplacian operators okay which say that the function H is a kind of S harmonic I call it S sub harmonic on these letters so I don't know there thanks for asking again indeed there is at least one result already so I uploaded the paper in the beginning of December and like in 10 days a result by marching appeared independently so so what I believe is is that it might be a good a good way to embed graphs okay that's the only point no theorems sorry apart from that but that's all