 Merci Nicolas pour l'introduction. Merci d'être venu. Donc mon travail ce matin, c'est de vous donner une introduction à ces objets. Et ce matin, vous allez avoir un peu plus de détails sur deux aspects de ces objets. Donc ce que je ne vais pas faire, pour exemple, ce matin, je ne vais pas discuter de l'approche historique de ces objets. Ce sera fait par Denis. Et je ne vais pas discuter de quelques aspects délicat des invariants conformés à ces observables. Ce sera quelque chose que, apparemment, Dima ne va pas faire ce matin, si je peux voir sa tête bleue et panique. Donc je vais définitivement discuter des invariants conformés aux objets. Mais je ne vais pas discuter du modèle éthique, ni de beaucoup de détails. C'est vraiment mon travail ce matin, parce que vous pouvez demander ce que je vais faire. Mon travail ce matin, c'est de essayer de prendre un exemple où la définition est très simple. Et aller par-dessus. Donc je vais définir le modèle, prendre le temps de définir le modèle. Et ensuite je vais définir le paramunique observable et vous donner un peu d'applications de ces objets en ce cas particulier. Donc je n'ai pas aimé une revue broad ou quelque chose comme ça. J'ai vraiment aimé un message de retour à la maison, que vous pouvez garder après ça. Donc, dans cette perspective, je vais définir et prendre un peu de temps de discuter le modèle que je vais étudier ce matin. Il s'appelle le Loop O N Model. C'est un modèle qui a été originalement introduit dans l'80s. D'un des authors, je pense, c'est celui qui l'a étudié le plus, c'était Nin House. Et son nom va venir dans le mix plusieurs fois. Donc le modèle est simple. Tu prends la lattice hexagonale. Tu parles une pièce, une pièce finite de la pièce. Tu prends Omega incluée dans la lattice hexagonale, qui est appelée H. Tu prends Omega finite. Et la configuration dans votre modèle. Je vais prendre quelque chose. Il y a une collection de non-intersectes loops en Omega. Donc eta est une configuration de loop. Simplement, si c'est un subgraph de Omega, c'est-à-dire que le degré à chaque vertex est Even. En la lattice hexagonale, il implique automatiquement que tu es juste une collection de non-intersectes loops. Donc, il y a une configuration de loop. Si c'est un subgraph Even Omega. Donc, c'est une collection de configurations. Maintenant, c'est la distribution, c'est la probabilité d'une configuration. Il y aura deux paramètres. Tu prends une X, une positive et une N. Normalement, ça devrait être plus ou moins de 2. Mais pour simplifier, il y a un peu de probabilités ici. Je ne veux pas les éclairer trop vite et les éclairer moi-même. Donc je vais prendre une N plus ou moins de 0. Et puis, simplement regardes la mesure. Je vais le appeler mu, en fonction de Omega, X et N. Et la configuration eta a une probabilité qui devrait être proportionnelle pour X à le nombre d'hégés en eta et pour le nombre de loops en eta. Donc, c'est le nombre de loops. C'est le nombre d'hégés. Si je le fais juste comme ça, ce n'est pas une mesure de probabilité. Je le renormalise pour obtenir une mesure de probabilité. Et cette constance ici, c'est la fonction partition de l'model. Vous verrez pourquoi je mets cet empty set comme un superscript dans une minute. Donc, ici, n'oubliez pas que c'est rédendant avec la définition de la configuration de loops. Mais ici, c'est la distribution qui est vraiment seulement sur la configuration de loops. Ok. Donc, ça me donne une configuration de loops en Omega. Je vais... peut-être avant que nous étudions ceci dans la généralité, je vais discuter quelques exemples spécifiques. Il y a un premier exemple, qui n'est pas très... peut-être spectaculaire comme ça, c'est si vous mettez n'est-ce que 0. Donc, si vous mettez n'est-ce que 0, vous pouvez penser ok, alors dans ce cas, la seule configuration que vous êtes regardant est la configuration empty. Ce n'est pas très intéressant comme c'est. Excepte que ici, vous pouvez facilement imaginer que j'ai restricté moi-même à même subgraph, mais je pouvais aussi regarder à même subgraph avec juste deux sources, comme deux places où j'ai un haut de degree. Imaginez que c'était ma mesure avec un set de sources empty. Je pouvais juste dire, ok, on va mettre deux sources. Donc, il y a deux vertices, A et B, qui ont un haut de degree. Si je fais ça, je peux exactement regarder la même chose. Et ici, je vais avoir une renormalisation, qui est différente, parce qu'il y a juste un set différente de configuration. Mais dans ce cas, n'est-ce que 0 fait parfait sens. Ça veut dire que je n'aurai pas de loops dans ma configuration. Ça veut dire que j'ai un path qui se termine de A à B. Donc, ok, j'ai en compte les choses bounderies, mais c'est vrai, ça peut être possible de faire des loops, mais c'est vrai. Donc, on va dire un haut de degree. Ça serait simple. Donc, n'est-ce que 0. C'est ce que nous appelons le model de mode de safe avoiding work. C'est juste un pass de haut, qui est de A à B. Il y a un autre cas d'interesse, qui est n'est-ce que 1. Et ici, c'est un peu plus subtle, mais parce que je vais essayer de donner un peu plus de détails. Donc ici, c'est la connexion pour le model de ease. Donc, fais le suivi. Donc, le model de ease est le signe de plus ou minus 1. Et ici, je vais assigner les faces de ma lattice hexagonale. Donc, la lattice triangulaire. Les faces et la probabilité de la configuration dépendra de deux paramètres. Il dépendra d'un paramètre. C'est-à-dire le beta. Donc, le signe va être appelé signe. La probabilité de la configuration de spin sera proportionnelle à l'expansion de minus beta h de signe divided by a renormalisation constant. Where edge of sigma is equal to minus sum of sigma x sigma y for x neighboring y. Ok, so let's call it another pair. X, y, an edge of the dual of omega. So, you have a dual graph and you are summing on pairs like that of neighboring faces. Ok? So, this is a classical model I think most of you already heard about it and you will hear more about it this afternoon so probably I'm not going to discuss this model in details. The only thing I want to say is the following. If I give you a spin configuration on the faces and if I start drawing all the edges between faces with different spins let's say I do something like that and then plusses and so on and I draw all the edges between faces with different spins so let's say this is one example what I obtain automatically is a loop configuration so it's exactly an object of this type and when I look at this expression exponential of minus beta times edge of sigma I can rewrite it very simply in terms of this loop configuration so there is a bijection let's say it's not completely a bijection it's a 2 to 1 map between sigma eta but the important thing is that exponential of minus beta edge of sigma I can rewrite it in terms of eta of sigma for the simple reason that an edge which is not in eta is between 2 faces with the same spin hence contribute here exponential of minus beta while every face which is between 2 different spins contributes there is a sigma x, sigma y which will be e to beta before sorry e to the minus beta so every face there contributes e to the beta every edge which is not there contributes e to the minus beta every edge here contributes e to the minus beta so I get e to the minus beta times the number of edges in omega which is a constant times e to the 2 beta and I made something wrong sorry it's something like that e to the minus 2 beta times the number of edges in eta of sigma so this can be re-expressed like that so when I plug that there I can really in the same way express the probability of eta as simply e to the minus 2 beta to the eta of sigma divided by a constant which is now not quite the same constant simply because I had to multiply by this but it's a constant so what did I do here I basically rewrote this thing in terms of this one in terms of this model so I cheated a little bit it's Monday morning we have a load why did I cheat it because it's not a one to one map right the reason is that if I give you eta of sigma once I give you the spin of one of the faces all the others but before that I have two choices just there is a plus minus symmetry so in fact here what I really want to do is to fix the spin somewhere so for instance you can decide to fix the spins outside of your graph so you just decide that all the faces outside of your graph receives spin plus and then it's a one to one map so here the true statement was in fact that I want to look at what we call the plus measure for the for the easing model so here you will have well let's write it like that but this time we let's keep it like that sorry but let's add minus sum of sigma x for x on the boundary of my graph so they all interact with one spin outside which is plus ok and when you do that and you write this thing you exactly end up here on mu omega x will be equal to e to the minus 2 beta n is equal to 1 there is no n to the number of loops in this picture et t of sigma so there is really it's an equivalent way of writing the easing model with plus boundary condition so n equal 1 x equal to e to the minus 2 beta this is just the easing model plus boundary condition on the triangular lattice at inverse temperature beta so fundamental example and this really I think that Dima is going to tell you much more about that probably not on the hexagonal lattice but it will be the same in particular there is one special case which is will be of interest for us in these lectures what is beta equal 0 easing model ok so it's an assignment of plus and minus 1 but there is no interaction because beta is equal to 0 there is no interaction between the faces so it's just coloring at random the faces of my triangular lattice in plus or minuses so it's called Bernoulli percolation in this case so n equal 1 equal 0 meaning x equal 1 well it's equivalent to Bernoulli percolation of parameter well I will color in pluses with probability 1,5 color in minuses with probability 1,5 so it's with parameter p equal 1,5 which is the critical point for this model so keep in mind these 3 examples n equal 0 n equal 1 general x easing model n equal 1,x equal 1 is just Bernoulli percolation so Bernoulli percolation sorry I should be more specific so Bernoulli percolation on a graph and here it's going to be site Bernoulli percolation actually site Bernoulli percolation it's just you color in every face black or white or if you prefer plus or minus completely independently for each face you toss a coin you color it black or white it gives you a coloring a random coloring of your graph and then what you are usually interested in there is the connectivity properties of this random coloring I will come back to that maybe I will tell you more about this special case anyway so you color maybe you color faces black or white so this is plus this is minus you need I mean a random and the probability of coloring black is probability p and probability of coloring white is 1 minus p so here it's p equal 1,5 Bernoulli percolation I will come back to that and tell you more I plan to do something about this model ok so pick your favorite example keep it in mind for what comes next so before I tell you about paraffeminique observable it's maybe a good point to tell you what I mean to think together about what are the good question on this model so first thing what you really want to do is to take omega larger and larger it's something which is not straight forward to do but let's even imagine you can take omega up to the whole lattice so you make omega go to infinity in some sense so the definition themselves don't really make sense because you need to be in finite volume to be able to count the number of edges and the number of loops but as I am guessing I will not surprise anybody that you can make just a limiting procedure and takes a weak limit of measures in finite volume to define something in infinite volume see something that people are very used to in statistical physics actually for the loop or in model it's a very difficult task because there is no monotonicity properties except in special cases but let's ignore that you can always anyway extract the sequential limit because the space of measure is tight so you could define measures in infinite volume in volume limit and you can study as soon as you have an infinite volume measure you want to study a family of infinite volume measures you want to study whether they are undergoing a phase transition or not so here what would it mean so phase transition you have different types of phase transition the most classical one is a phase transition between an order and a disordered phase so here maybe the equivalent would be to think maybe there is a range of parameters where the loops are all small and well small in the sense for instance that the biggest one I will encounter I mean they are never like bounded size uniformly in the lattice so maybe the biggest claim you can hope is that the biggest one you will see at distance n will be logarithmic so that will correspond so it will be corresponding to an exponential decay phase where basically the probability so this is the definition mu of hxn of say the loop of vertex a has length larger or equal to k is decaying exponentially fast in k so in this phase because there are polynomial number of vertices at distance n the biggest one I will see the biggest loop I will see will typically be logarithmic so that is one possible behavior of this infinite volume measures another possible behavior will be that there will be an ordered phase in the sense there will be for instance an infinite loop crossing the lattice this in fact will never occur you really have to think that we are in costelis tau less type phase transition for the physicist in the room you will never have an ordered phase so what is alternative behavior so for people for probabilities who are used for instance to percolation you have this disordered phase where all the clusters are finite and have exponential tails you have this phase where you have an infinite cluster of plusses so there are only two phases well not really there is one phase exactly at the critical point where the behavior of the clusters is polynomial I mean you have positive probability of having big cluster and the probability that the cluster at the origin is polynomially big is decaying polynomially there is only one point with this occur it's a critical point well here there will be a whole range of parameters at which the loops will in fact decay polynomially so I will call this a critical phase these are not standard denominations but I think since it's an introduction it will be sufficiently intuitive for everybody so in this phase the probability that the loop at the vertex A has length larger or equal to k will be larger or equal to 1 over k to some constant c I will put a small c here and big c here ok so in this phase you can actually restate the thing in the following you respect the following so that's I mean this is really the interesting phase so in this phase many things happen and in particular if I look at a ball of size n around the origin in this phase there will be actually macroscopic loops there will be one loop which is one or more but there will be loops which will roughly have the size of the whole box and in fact I will not have only one I will have a family of loops maybe one which is twice smaller some like that etc I will have in fact a compact in some sense a tight family of random loops so a natural question if in a box of size n you typically see something like that a natural question is what happens when I rescale my lattice so that my box exactly fits in a box of size 1 and take a look at the random family of loops that I obtain as a family, as a limit when n tends to infinity so just rescale imagine this is but the hexagons have size 1 over n ah, classical yes and I also added eta because I want to that you are very careful so let's put delta and this was 1 over delta so now it's 1 so you take hexagon of size delta let's define mu delta to be the push forward of the measure by this rescaling so it's just a measure that you will obtain on that then you have a random collection of loops so it's not completely clear what you mean by that because you have more and more loops but for instance what you could imagine is to cut only the loops larger than epsilon say, epsilon is a threshold and take the limit and the question is what is this limit what is the limit of eta delta which is simple according to mu delta so what is the limit of this family of loops which has low mu delta and in this regime so by the way in the other regime here if you do this limit and for every epsilon so you fix epsilon you look at only loops of size larger than epsilon well in this regime because the biggest one was logarithmic the biggest one here is logarithm divided by, I mean multiplied by delta so it's log of 1 over delta times delta so it will just collapse all the loops will collapse to points they will never have size larger than epsilon so it's empty an empty configuration in this regime in this one because you have positive probability of having big loops you end up with something non trivial and this something non trivial instantially non trivial in the sense that physicists expect that in addition it has a lot of symmetries so here maybe that's a very poor drawing of a box of size n on the hexagonal lattice so a box of size n in the hexagonal lattice has a shape like that so what do I mean by a lot of symmetries so imagine eta delta converge as delta tends to 0 so for instance in low to collection of loops in the continuum let's call it eta like that well this guy the low of this guy this guy obviously is invariant by rotation by pi over 3 2 pi over 3 sorry simply because the original measure was already invariant under this but what physicists predict is that in fact this measure has much this object has much more symmetries many many more symmetries in fact it's symmetric under any conformal map meaning that if I take so here I took one box an hexagonal box but at the end you agree with me that I could have taken any shape and take the limit of this object this is the boundary of my box well there are conformal maps mapping this to this and in this new domain there are two natural objects that you can define if you have a conformal map from this to this let's call it phi here you can just define eta so let's call this guy omega eta omega just by taking this limit just I take this domain, I take the same limit so the scaling limit but I could also take the image of the limit I obtain in this domain in this one, I'm cheating a little bit because here there are three parameters that I'm swiffling under the carpet but I mean it's not an introduction to conformal theory it's an introduction to Paraphymedic observable so I want to dive as fast as possible on the I mean in the heart of the subject but here what I claim is that I can just take the limit of these loops there map it by my conformal map phi, I end up with a family of loops here just it's the image of loops there and what I claim is that these two objects they have the same law that's what is predicted so conformal invariance means that these two objects have the same law so we will hear I think more about this afternoon so maybe I don't tell you too much about it but what I wanted just to emphasize is that in this critical regime there is a very I mean rich class of possible behavior for the models for these loops and there is a very rich scaling limit ok and the natural question is how can we understand this can we get a grasp at this which are for instance the parameters for which I have this behavior for which parameters do I have this behavior if I am in this behavior can I prove for instance conformal invariance that's a big program right don't worry the answer is no in most cases well I will prove conformal invariance of something today but you cannot do it in full generality ok so at least now you understand where we want to aim or at least I believe you understand where we want to aim ok so in order to do that we are going to introduce an object it's going to look maybe a little bit mysterious but I want to leave some yes yes of course is there some implicit claim that there is a unique choice of x and n for which you have this phase this conformal map, this phase so I will come to that yes I will exactly come to that well let me come to that but it's indeed a very natural question for which parameters do you expect to have this phase and in fact I will come to the fact that there are many ways of being conformal invariant and even just identifying which way is already a challenge it will not be independent of x and n it will be independent of omega yes yes for certain parameters of x and n it will be conformal invariant meaning for any omega I mean for any omega and for any conformal map from omega to another domain omega prime you will have that eta omega and the eta omega prime and the image of eta omega by the conformal map are the same law do you view these two configurations as configurations in full plane no really in the domain so if you cut here if you only look at the loops in this and you take the image you obtain that so I mean here I was a little bit not precise on the fact that I define this on the full plane here in fact to really claim this conformal invariant what you really want to do is a measure which is really here you take loops and only in this domain so you take really mu omega xn you take the limit so by definition you have no loops that exit the domain because you are really in this domain there you do the same with this other domain which I so here it's mu after is scaling it's really with double bar there it's really the infinite volume but how can you compare the laws so then they are just families of loops on the same domain right so these are at least the state I mean the space of configuration is the same here they are different here they are on this domain but I really conformal map by phi right so then it's a family of loops on this domain sorry? so they are random objects right so I only claim that they have the same law and indeed here for a given configuration really do not hesitate to ask questions for a given configuration it's not symmetric right it's just the law of the object is symmetric the law of getting that hexagonal graph is symmetric but if I just intersect this domain I would get something that looks very different even combinatorially exactly yes that is true and that's where physicists are better than mathematicians because the mathematician would be scared by that I think rightfully but physicists are not so the point is that there indeed the lattice is not at all symmetric by anything the lattice is really this one but you take the limit first so you cannot forget in some sense the lattice and what the physicists are telling you is that you really forget for instance the orientation of the lattice you started with but this is something beautiful and miraculous in some sense it's not miraculous but surprising at least the first time you see so here the symmetry by 2 pi over 3 is really obvious from the model for instance even just the symmetry by 2 pi over 3 is not at all obvious they do because you conformally map here these are loops on this domain but you conformally map by phi phi is a map from this domain to this one so then any loop which is there but phi of this lattice is not that lattice sorry yes but this is not important because here it's infinite if you want it's really here let's call it 0 plus it's really the limit when delta tends to 0 so in the limit there is no reason you don't gain additional symmetries just think simple random work on the square lattice you move your random work on the lattice really feels that there are four cardinal directions right you take the limit of these objects you end up with Brunian motion which is completely rotationally symmetric so you gain symmetry in the limit that's the whole beauty of the whole theory here it's exactly the same that happens you have macroscopic symmetries but not microscopic symmetries yes it's not obvious at least at the level of the lattice I mean with our theories it's absolutely not clear how you end up with this additional symmetry so already the rotation yeah even assuming and this is the question I like because I think there are ways for instance for the random cluster model to kind of sense that for any of these models you have rotational symmetry but I have absolutely no clue how you would deduce from that and scale invariance yeah that's sorry exactly you should add locality which is really not so physics there is a folklore that is locality yes that is true but ok so in order to try to understand that let's try to introduce a model a tool which will look a little bit mysterious this morning but we will have the deep roots of this object this afternoon we will have some perspective on it so I will kind of keep the mystery and just define the object and work with it so these objects are called paraphernes uniques observables so so really if you are a little bit confused or if you are not convinced of the invariant speech just think of the questions do I am in this phase or in this other phase that's already a sufficiently hard question for us anyway ok so the idea is the following you are still expecting conformal invariance so you are expecting that your loops have a lot of symmetries and these symmetries are related to conformal maps so the game is going to be that if you have that then you expect that certain just observables certain random variables expectation of random variables have the same symmetries ok then this next step is to think ok if you have this can I find can I guess which observable they are and if they have symmetries conformal symmetries they are probably conformal maps so can I actually guess in the discrete model which is on a lattice where indeed there are cardinal direction direction that are preferred by the lattice can I find objects that already are some sense of finite volume versions of these very symmetric observables ok and object will be the following so the first guess will be well I want a function that depends on a point so what I could try to do is to introduce a singularity introduce a place where I have a odd degree in my model instead of even ok so I could try to put so this is now omega it's really in finite volume forget all this taking the scaling we will work in finite volume so take this domain omega and take a point inside which I would call z and I want z to be I will evaluate certain things at z and this would be a I will see it as a function of z so I want that at z for instance I have odd degree so there will be a path starting from z and loops so because I need necessarily to have an even number of sources I need another source so I could try to put another source inside I just need that and then I am yours so you could put another source inside what I will do to simplify a little bit the matter I will put my source on the boundary ok so I will put another source here on the boundary I will call it a ok you have a question exactly so I answer and then I go back to it's not at all obvious by the hexagonal lab displays especially for yeah I will come to that I will really come to that second question will you ever actually tell us what we now know about these neon eyes or when models I will also come to that excellent we are on the same yes no ok so so my goal is to have pictures of this type so first I want to be a little bit precise on what I am going to do is z and a will be middle of edges so you have an edge here and z is the middle of an edge you will understand why it will become important later and same thing for a on the boundary so what I will do is that the boundary if you think about it looks like that right it looks like that oh yes it's a terrible choice ok it doesn't look like that but I think chemically it does oh that's a I will never learn how to draw this thing so actually what I mean a boundary vertex is I really took a half edge exiting my lattice like that and I pixel say this is a and I have a pass like that going from a to z so this is now my space of configuration a is fixed z think of it as a point which is like the variable in the function I will define ok so the most natural function you could try to define in this case is the following so you could try to define z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z 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