 So the subject of this talk is not directly related to random matrices, I will spend some time defining the setup of it. So the model I'm going to talk about is called the random walk in random environment. Then we spend a little bit of time defining it. So this is a stochastic process which corresponds to the movement of a particle on the hypercubic lattice. Well, it could be some other graph, but I will talk about it, I will focus on this graph on the hypercubic lattice. And the particle has jumps, for the sake of this talk, the jumps will be always two nearest neighbors, so the possible jumps will always be some unit vector in the lattice. So this is the L1 norm. And now the jumps of the process of this random walk are itself random. So this means that the jump probabilities will be random, and they will correspond to choosing at each side some set of numbers. So these numbers are the probabilities to jump in a given direction indexed by the jumps, so they have to be positive, or they will not be probability, they have to sum up to one. And this is what will define what I'm calling an environment in the title. So actually I will define what I will call the environmental space. This is just means setting at each side, or prescribing at each side some jump probability, which I am defining as this Cartesian product here. And what will we call the environment is just choosing, as I said, these probabilities. So at each site I have to make a choice. So this will be some element of the environmental space. And so the omegas are a set of numbers which are in this class of numbers which sum up to one and which are positive. So you can, okay, these are definitions which in a sense it's much easier to understand them making a picture here. So what I just said in all these definitions is that if I have a site, X, let's say on the lattice, and I want to know what's the probability that the random walk will jump from site X to X plus E, this will be given by the environment that I choose, and will be given by the number omega X E, that's what I'm saying. So actually the random walk itself will be defined by its transition probabilities. So for example, I will fix the starting, this is going to be the starting point, and the environment, and the random walk will have a certain probability of jumping from a site Y, let's say at a time n to a site Y plus E, which is just given by the environment. So this is for all points on the lattice. As you can see, the process I'm looking at in this talk will be discrete time, it's not crucial, and the process is starting always from site X in this case. So well, this is a Markovian process if you fix the environment at least, but what I want to do now is I want to put some randomness on this environment. So that, of course, is just given by, will be modeled by putting some probability measure defined, probability measure in the environment. So I will call it P, the double bar here means that I will distinguish the probability measure on the environment, which is going to be written by double bar, I say P with double bar, sorry, from the law of the random walk itself. So there are usually two ways of looking at the random walk. So one is, the first one is what's usually called the quenched, let's say quenched point of view, which means that the environment is frozen. So the law of this random walk I already introduced it is going to be called, I will call it P with a subscript X standing for the starting point and omega for the environment. So, but there's another point of view which is the, it's called usually the averaged, sometimes annealed, which means that somehow I want to study the movement of the random walk, but averaging what's going on with respect to the environment. So I'm just essentially taking an average here over the environment of the law. Both, the behavior of the random walk under both measures is not necessarily the same actually. And actually the second, the average law has the disadvantage that the random walk is not Markovian under it. Okay, so there is a behavior which is going to be important to distinguish throughout this talk and maybe I will make some assumptions right now. So the first one is that I will assume that the choice of the jump probabilities is IID under the law. And the second one is that I will suppose that all of these probabilities are larger than some constant. I cannot have any jump probability which is equal to zero. So I need to introduce a behavior which will be relevant in my discussion which is what's called the ballistic behavior of the random walk. So I'm going to introduce a direction. The idea is that I have some direction here. And there is a possible behavior of the random walk which is called transient or transience in direction L which means that the random walk somehow wanders a way to infinity in direction L. This point here is just the inner product or the projection of the position there on the walking direction L. So in the picture it would mean that you're on the walk and you move backwards somehow but at some point it will end up going to infinity, it could oscillate but its projection in this direction will go to infinity. The other behavior which will be relevant is what I already said is ballistic in direction L. So this means that the transient behavior occurs with a given speed so this is a much stronger statement than transience or transience, directional transience let's say. So well there's an open question which I'm not going to discuss in this talk I'll just say it which is that it's believed that or many people believe that in dimensions larger than one any walk which is transient in a given direction is necessarily ballistic. So that's still open. I mean I wouldn't bet on it actually but okay so I'm more interested in scaling limits of the random walk in this talk now so I need to go on to large deviations for people to understand or to prove let's say a large deviation principle for the random walk and random environment. In dimension one it was proven in the 90s but at least for any dimension it was only in 2004 that it was proven by Varadan and well he proved two things first he proved that if the law of the well I said that I was going to assume that the law is IED but it is the only exception in this point that I'm going to say something for a law which is not IED necessarily. So if P is stationary and ergodic then so Varadan proved that a quench the large deviation principle is satisfied so I don't have time I'm not going to define what a large deviation principle is I think everybody knows should know or knows but let me just say that in in particular one can one one can show that in this is not completely trivial the corollary that if I look at the probability at the quench probability that the random walk in is at a position xn which so which asymptotically converges to a point when you normalize it because the range of the random walk at time n is a l1 ball of l1 norm of radius n so if you look at if if if you look at this quantity this converges actually to the rate function at x so this is the rate this is the what's called the quenched rate function it's not random and he also was able to prove a an average large large deviation principle so analogous in an analogy with the limit I just wrote one can show that his average large deviation principle implies now that if you look at the same expression but under the average law this converges to minus another number which depends on x another function let's say which is the average rate function so both of these rate functions are in not too much is known about them very little let me say some of the things so this is a very basic property turns out that these functions are finite this is very simple to if and only if in you look at points in an RD which is I said already which have a l1 norm smaller than equal to one the same is true for the quenched so this this set D will be important in the rest of the talk in can imagine it as a diamond since okay or a square which is rotated and now in general these rate functions are not equal the annealed and the quenched rate functions are not equal in general but a for example for example in dimension one they are they are in dimension two it's known that they are different but there's a result of the 2010 of you must he was a student of Iran who proved that in there is a if if the random walk is ballistic so this is what I already discussed at the beginning well he assumes something not acceptable necessity but then in there is an open set where the rate functions coincide I'm not saying I'm not giving you all the information of the set let me give a little bit more in this picture the set the set D is the open set which is here inside is in is a set which contains an important point which is the velocity when the run the walk is ballistic the velocity is not zero and this open set you can imagine it with something like that this is the only thing which is known but he has assumed this very somehow I would say it's an annoying thing which is ballisticity I don't have time to explain why but it's just somehow ballisticity it gives a structure of the law to run the walk the annealed law which is where you can you can see some IED structure of the run the walk and the other there's another result of 2008 this is not Polish but in who was also a student of rather and I'm not even going to state that I'm just going to say that he gives a variational formula for the quench rate function and it is very non non explicit but say somehow an abstract variational formula which I would say has not been very useful conceptually it is interesting but it has not been so usual to to make computations in a sense of the of these rate functions so somehow the big problem is how if somebody could compute or give some formula for these nice formula for these race functions okay so the so recently with with other people so the Rodrigo Bassais and Chiranjeev Mukherjee and and we were able to in a sense get rid of this assumption but there is a cost which is very natural and somehow gives the right I would say the setting for the quality of the rate functions which is low disorder and I forgot to say the result of the mass is the valid only dimension larger than equal to four otherwise it's not true so here's all you will also you also need dimension larger than equal to four and a low disorder so I will define it right now so there is a you can define for given a measure a law of the environment p you can define it's disorder measured as in the maximal distance between the random probabilities at a given site with the their averages so this this q is the average of omega so this a number this epsilon is measuring how far the the maximum the the maximal distance that you can have between the randomness omega and the q so the theorem says that there is a for for any compact set which is in the boundary of this diamond so I'm looking at the boundary not only now there is an epsilon in an epsilon which depends on on the compact set such that in if the the disorder of the environment is smaller than this epsilon the rate functions agree on the compact set the the the first important point of the theorem is that a it gives the equality of the rate function without the assumption of elasticity if for the moment the the result is only in the boundary in the interior with it is work in progress and as a corollary the you you can you have the an explicit rate an explicit sorry formula for the for the both rate functions so this is a because the anneal rate function at the boundary is can be computed immediately so you get a formula for a formula for the quenched one and I used to say well that the the there is really a phase transition here it is possible to show and I'm not going to say the the whole the the full all the details of the theorem that in general if epsilon is large enough these rate functions are not going to be equal and okay now the proof I I I don't I want to I mean that's not the only thing I want to say in this talk so I don't have too much time I just want to say about the proof that a in one a observation that is important about the proof is that in there is a martingale in which plays a role here which is what I'm writing over here if you're able to get a good estimate on on a norm of on the L2 norm of this martingale in it is it is possible to show that in that actually the martingale this this this thing is converging to zero and this will give you the equality of the Laplace transforms of the the random walk and then you use a you you can use a very well known theorem of Gartner and Alice to get the the equality of the rate functions okay so now let me and go on to what I wanted to say about what happens in dimension two and so in dimension two you you cannot expect a result like this or you have equality of the rate functions even if the disorder is is is low but but actually and it is possible to find out this I want to talk about a make a connection with here with a what's called the kpz universality class okay this this this is a a somehow a picture of a universal behavior describing the movement of interfaces two dimensional interfaces which separate a stable phase by from an unstable one and so it turns out that the random walk is in this class also I want to I'm not going to convince you because I'm not I'm not going to have time to to give you the proof but at least give you the statement and so we'll have I will have to be a little bit more precise about the disorder this this rises I mean this this connection arises when one looks at the random walk at low the sort again but we will look at environments which are of this kind so the sort of appears in a particular way there's a parameter epsilon and and these size are iad again so you have a set of the laws which are which are joined they have a joint distribution given by the size and they are parametrized by epsilon but for the effect of the scaling I will do I will put a square root of epsilon here it's just a convention not it's not a and so okay I will make a scaling and yes I didn't say this the these size are also bounded by one otherwise okay so we'll make a a scaling which is not so simple to define so first I will define I will I will I will choose some some point on the on this space where the rate functions are finite and please I think I still have 80 minutes or or nine maybe nine and so let's make a picture here I have the I think I'm not on the blind spot I hope so I will choose some some point here as I said and now the second ingredient I need is a I will I will have to define some curves which have a well this this equation is is I don't know if it somehow this is an equation which interpolates between the equation of a circle and the equation of a square so for for small in if if you rotate the coordinates this corresponds as I said to the equation of a square for the constant a large constants so for for small values of the constant this is very close to a circle but as you increase the value of the constant you begin to to look more more like like a square so here even it's like this and and in the limit it will really be the square so these these are these these curves in a constant so now I will choose maybe maybe for the sake of the explanation I need to choose this point somewhere else like that let's say w I will choose a a the unit tangent of the point I'm I'm pointing at the which is a tangent in w passing through the curve so this is unit tangent so what I what I'm going to do now is it's a scaling in where time the time of the random walk is going to scale like like well diffusively in a sense like epsilon square but space will have two way two coordinates of scaling one is the omega the omega the w one sorry the w one is going to scale like time so in a sense this is w is going to be played the roll of time and the orthogonal one will scale like like its square root and well of course noise as I defined the model is scaling like the square root so now I think yeah let me see the theorem so this is a joint work with with well Jeremy Quastel so the the theorem says the following it says that in well dimension two there is a there is a well there are some functions which I'm going to call the first one alpha sub epsilon alpha sub epsilon is something that will depend on the on the point which I chose here on on a time I have to fix time I have to fix space this is this x is one dimensional and there's another scaling normal or you say normalizing function which also that will depend on this on the three quantities and also a couple of constants these ones depend only on on the direction I choose so what's going on is that if you look at the transition probability this it's the same transition probability I was looking at before the quench the one sorry the probability that you run the walk at time in is let's sorry I should write here at time t over epsilon square I'm not saying of course you have to take the integer part otherwise this is not defined I'm looking at the street time and so you as I said there you rescales one directional space like time but the other so this is a vector remember the other direction like the square root and the result is that this converges to and you have to center by this landa this converges in distribution to a function which is it is a stochastic function which is the solution of the stochastic heat equation and so where as I said let me write it this is the solution of the stochastic heat equation and there's the the gamma here u and and then you have here well maybe I should write it like that is space time white noise so why I said that this gives a connection with the kbz equation because the the the kbz equation is the help called transformer of the solution of the stochastic heat equations in a sense this is giving the relationship I want to I want to point out that the result is it's what's relevant about actually a theorem I didn't say is that the it is a result about essentially a model which is a polymer which is not directed and most of the well I think all of the results giving the scaling to this stochastic heat equation were under the assumption that we have some directed model and maybe I should just say very briefly that there is a I didn't I didn't speak about this today but there is is there a worker problem is the worker problems with my friend the Jose Ramirez it's not Alejandro Ramirez where we we can apparently there for a certain choice of the law of the round the walk and round the environment you you you can find determinant formulas it has to do with the talk that the first class I'm not going to say more about that okay thank you