 And then if there's time at the end, I'll say something about where they're coming from. So first I'll describe the doob-transformed walk. And then the wandering exponent result, two-thirds. And then finally a connection with large deviations. All right. So let me just start stating some precise results. So now, by the way, when we're in this beta situation, let's record this. The mean of a beta random variable is alpha over alpha plus beta. So the law of large numbers velocity c star is now alpha over alpha plus beta, comma, beta over alpha plus beta. So that's kind of good to keep in mind. I have a feeling this thing keeps wanting to fly off. I guess I don't quite know how to use it properly. So what is a doob transformation? I should tell you. So the general fact is the following. That if you have a Markov transition on a state space and a function on the state space, that's harmonic. So if H is harmonic for a Markov transition P, so this means that pH is H, then you can define a new Markov transition probability, pH of X, Y by doing this. You go PXY times H of Y divided by H of X. So the harmonicity of H makes this a legitimate Markov transition probability. And let's assume that H is positive, so it's perfectly okay to divide by. So that's a doob H transform. So you need a harmonic function to do it. So where's that going to come from? Well, that comes from a limit, which is kind of an analog of a Boozeman function for this model. And let me not clarify that technical term even though one should. It says something to those who know what I'm talking about, but otherwise it's not that important. So here's the first theorem. So fix now XC, which is one of the admissible velocities, but not one of the endpoints. So I wrote relative interior of U. So the picture was this again. U was the simplex where all the legitimate asymptotic velocities lie. And relative interior is all of it except the endpoints. So fix one of the XCs. And then the statement is that on the space where those weights live, so that description is gone now, but it's the weights omega. On omega, there is a stochastic process. And I'll denote by B superscript XC, which is sort of a label. And then the process itself is indexed by lattice points, x, y. And this process has the following properties. So let's say such that. First of all, if I take e to the negative B, 0x, then this is harmonic. So as a function of x. So let me write explicitly what that means. So it means that if I start from x and I look at the average of this function e to the minus B going one step, then that average is the same as the function at the place where I started. So transition from x to x plus e1 times the function plus the other transition equals the function at the starting place. So it's harmonic. I should say so omega is all the time present here. So this is for almost every omega. Everything is for almost every omega. So there's a hidden omega here. This is a process. So it's a function of omega. I'm just not writing it there. So there's a global harmonic function like this. And it's coming from certain limits. So this is the second part of that theorem. So 2, this Bcxy at omega is the limit as n goes to infinity of the following difference. Log quenched probability starting at x and then going to a point zn. So I'm going to write like this. So I want the probability that my path goes through a particular point zn. So if I start at x, I need to take the appropriate amount number of steps to get to zn. I don't want to figure out what that is. So I'll just put x dot there. So saying that the path x dot contains the point zn. So this starts at x. That's the same x as there. And then I take minus the same probability starting at y like this. So a limit of the differences of the logs of those two quenched probabilities for any sequence zn such that zn over n goes in that direction c. So there's the c here that was fixed at the outset. So and this is again bold P almost surely. So let me say it again in words precisely. There is a probability one set of environments omega such that no matter which sequence you pick that satisfies this limiting behavior, this convergence holds to that value there. And maybe a third property is that what if you pick c equal to the law of large numbers velocity? Well, then you just get identically zero. So that's the first part that these limits exist. And they define a harmonic function. More can be said. So let me add maybe this bit. It's possible to say, so this b, c process, this has certain properties. You can see from here that there's a certain additivity. If you go from x to y plus you go from y to z, that's like going from x to z. There's also a stationarity here. If you translate the omega, then that's the same as translating the x and the y. So it's a stationary process or has stationary increments. And we can write down explicitly what the distributions are. So the, let me actually cheat from my notes so that I don't get this wrong. Not that it would really matter to any of you if I did, right? So e to the e1 increment is a beta variable with parameters alpha plus a certain function lambda of c and then the beta. So now I guess I don't have the, remember, our hypothesis was now that our initial weights were betas with parameters alpha and beta. So here we pick up a different parameter with a function lambda of c. And then for the other increment, the e2 increment, it's a reciprocal of a beta variable. The exponential is a reciprocal of a beta variable. And this is to be precise lambda of c alpha. So lambda of c here is an explicit function. Can't quite give a closed form for it, but it's defined by an equation that involves some trigama functions. So this is pretty explicit, this lambda of c. So the fact that we're able to do these kinds of calculations, now that's of course entirely due to the fact that we have the beta environment. It has some special properties that enable us to do something this precise. Unfortunately, I can't really go into those properties because I think I'm down to 10 minutes or so, but that's what's going on behind here. But now I have a harmonic function and I can use it to do transform the original transition probability. So I'll define a new transition probability. Let me call that pi super c. So it's associated, oh, this here was a c, sorry. So for each c, there's one of these processes, right? So for each c, I can now define a new transition probability, pi c, to go from x to x plus ei. So it's going to be, so now it's going to follow the Dubb recipe over there. So it's going to be the original transition probability of omega and then multiplied by a ratio of h's. What's my h now? My h is now this harmonic function here, e to the minus b. But because of the additivity, I can combine, I can combine that ratio into a single exponential. So e to the minus b c of x to x plus ei. So that's now a legitimate transition probability. And it respects shifts again because these two ingredients do. So it's a legitimate RWE. But now you see it's, of course, highly correlated because these b guys, as you see, they depend very globally on the environment, right all the way to the asymptotics. So this is now a very correlated transition probability. And let me denote by capital P of pi c, the path measure that obeys these transitions. So path measure with transitions pi of c. So I have an RWE now, but in a very correlated environment. What sorts of properties does it satisfy? Well, this transformation that we now did, one of the things it did, it changed the law of large numbers velocity, which seems to have been erased also. It changed the law of large numbers velocity from the original c star to this c. So that only tells us that this new measure is singular to the old one. So for here's the statement, for P almost every omega under the new transitions, this c is the new law of large numbers, new law of large numbers velocity. Okay? But this is now the walk under which we will see the wandering exponent. Not under the quench, though. We expect that under the quench, we will see some kind of very dramatic concentration, like for directed polymers. And we're sort of working on proving that, but I don't have anything yet to report. So to see the wandering exponent, I'm now going to take average of omega once more. So let me write P bar with a starting state and a c for the path probability that I get when I take this doob-transformed walk and average out the omega. The omega is here, right? It's still a quenched measure, a function of omega. So let me put maybe a dot here. It's a measure. So average out the omega that's sitting up here. Under this path measure, we see the two-thirds exponent. So let me state a theorem to that effect. Let's see what's good to erase. Maybe I'll keep that so far. This is just general blah, blah, blah here. Solve theorem. Notice, let me draw your attention to something here. When xi is xi star, we don't see anything new. Cb is identically zero. So when xi is xi star, this is just one. So you have the original transition. So you're not seeing anything there. So now I need to assume that xi is not xi star and then we can see something interesting. And here's a precise state. So then we have constants, big xi and little xi, positive finite, so that we have the following kinds of bounds. Let me just look here again. So there's an upper bound and a lower bound. So for the upper bound, we have the following that... So I take this averaged measure here and I ask that the walk deviate from nc by an amount b times n to the 2 thirds. And that's going to be bounded above by capital C over b cubed. So let's parse that for a few seconds. What is it saying? It's saying that if you ask to deviate by some large number b... Here we go again. You ask to deviate by some large number b times n to the 2 thirds, then the probability is small. So this says that the fluctuations are not larger than n to the 2 thirds. And then from the other side, here's a slightly different kind of statement. Take the expectation of the deviation, then that is at least this little c times n to the 2 thirds. So those two statements are different, but they do together... They say that the typical order of fluctuations is n to the 2 thirds. So here's something that's also typical to this KPZ line of work. We have these fluctuation exponents as sort of... They don't have precise mathematical definitions. They have vague, qualitative definitions that they're orders of magnitude of fluctuations. And then the actual mathematical definitions of the exponents are nailed down once people can prove something. It's what you can prove, then that sort of determines how you define the exponents. So that's the wandering exponent result. I guess Ivan, when should I stop? 55 or... Sorry, 2 minutes, perfect, yeah. So what to say at this point? Maybe I'll just say... So this path measure was kind of introduced by Fiat now by doing this dub transformation. And if you still remember back at the first few moments of my talk, I did say that it's also... It's the random walk conditioned on an atypical velocity. So let me add that theorem for the last thing here. That this transformed walk comes as a limit when you conditioned the original walk to go in that direction c. So again, take a sequence of points going in direction c. Then the statement is that if you take the original quenched path measure, the very first one that I define in the IID environment and you condition it to go into this direction c, then this probability measure converges weekly on the path space to that dub transform measure there. And this is again for almost every omega. For almost every omega independently of that sequence. In other words, there's one event of probability 1 such that whatever sequence you pick, you get this convergence here. So this is really already contained in that earlier limit statement. It's an immediate consequence of that. But it illustrates that this is sort of a natural thing, this dub transport measure. And this family of transport measures is sort of closed under this conditioning now. In other words, if I take this guy now and I condition that to go into another direction, not that same c but something else, anything, I get the corresponding thing with the pi zeta or whatever up there. So in that sense, it all fits nicely together. Okay, I think that's a good place to stop. So thanks very much for your attention. Yeah, so all these solvable models, the Taysep that Jeremy talks about, and this one and the handful of others we know, they always have some kind of a miracle that happens. And this miracle, well, there's several of them, but they always seem to go together. There are combinatorial miracles and some probabilistic miracles. So let me try to say this in a few minutes here. So harmonic functions, where do they come from? By solving Dirichlet problems. So here's what one can do with the beta. Fix a point v here on the lattice. And imagine starting the walks at x, any x here in this quadrant, and letting them run and hit this boundary up here. So by solving a Dirichlet problem, what you do is you put some boundary, you put some function rho here on the boundary, and then you extend it into the quadrant by defining that rho x is the expectation starting at x of the rho value that you read when you hit the boundary. This produces harmonic functions. Now, with the beta environment, it is possible to specify some functions, boundary functions, rho, depending on omega, that again, as functions of omega, have nice beta distributions along these increments and so that this solution here is tractable. By that, I mean that, see, everything is a function of omega now. This rho is a function of omega. Quenched probabilities are functions of omega. This guy here is a function of omega. By judiciously choosing these boundary values, I can produce harmonic functions whose distributions I understand very precisely. And that's where the miracle of the beta comes in. And in fact, it's right here. So on the boundaries of this quadrant, I choose increments here in the E1 direction according to this distribution and I choose increments here in the vertical direction according to, well, the reciprocal of this distribution. And then when I solve for my harmonic function, I get a function whose increments down here in the bulk have those same distributions. So, and I have a whole family of them because I can vary that parameter lambda. Now I can use, now I have something to calculate with and then I can start, for example, trying to constrain those these kinds of probabilities here. So I can use, this picture is informative. When the walk starts from here, take this to be NC. Now, here when I try to control these deviations, I'm basically controlling how far from this corner the walk hits. And I can perform some of that core control with these harmonic functions whose distributions I understand. So that's sort of not so what's behind the proofs. So probably we don't have time for the questions for that.