 So now I'd like to talk about how you can really say anything about these processes. How can we produce these somewhat, you know, unusual Markovian projections? And for that, there's a framework of Paul and Schoenlingkopf. And it's called the framework of intertwining diffusion. So let me describe this framework. The idea is that you abstractly start with two diffusion processes, X and Y. And what you ask for is conditions under which a third process, Z, will couple both of them. So our setting Y will be the Laguerre eigenvalues process at a certain level and X will be the process that we've constructed at all lower levels. And what we want to find is a process Z, which would be the Laguerre-Warren process, that couples them and has all the properties we want. Okay, so let's move to this more abstract setting. We just have two processes with these generators, and they're going to live on a certain domain, which in our case will just be the Gelfand-Selin code. Now what I want is that there's some sort of notion of Gibbs restriction from measures on the top level to measures on the lower level. And that's going to be implemented by this Gibbs density lambda. And in their framework, it's called a link function. Okay, so what they define is that a process Z is an intertwining of the diffusions X and Y, essentially if it has the properties that we want. So in particular, if I project to the first coordinate, I should recover X. Project the second coordinate, I should recover Y. That means that I have a coupling of X and Y. Second, what I want is that the initial conditions are Gibbs, and actually all conditions are Gibbs. So that's properties one and four. The second thing I want is that actually provides a coupling. So that's property two, which is just the condition for two Markov process to become. And finally, what I want is that the process X somehow evolves independently. So really all the coupling is happening with the process one. And this is going to correspond to the fact that if you look at the Laguerre-Warren process, the first K levels evolve independently of the K plus first level. So really all the action is happening at the top level, which would be Y here. This is actually quite similar. It's a bit more restrictive. And the upshot is that we're going to give a theorem which allows for construction of these intertwining diffusions under some weaker conditions. So in particular, condition two is very similar to this Piment condition. So what we're going to do is give a general construction of a Z which intertwines X and Y under certain conditions. So it's going to be presented as a solution to a certain stochastic differential equation with reflection. So here's the equation. First, for Z1, the equation is just simply the exact same as the equation for X. And that corresponds to the fact that X and Z1 should evolve in the same way. Now for Z2, the second coordinate, I take the same driving terms as for Y and then I add this correction term, some drift, which somehow should correspond to the push from reflection off X. And third, I have actual reflection off X. Okay, so this is some complicated stochastic differential equation with three terms. So this corresponds to the coupling with X and this is some term which enforces reflection off X. And perhaps you can see it more clearly, the generator for this diffusion Z is simply that I evolve X, I evolve Y, and then I evolve Y a little bit more because I'm trying to couple the two processes. Okay, and I'm going to want a special property on this process as a diffusion. What I'm going to want is that, well first of all, its domain always contains these functions. So these are compactly supported functions which obey a certain Neumann condition. And the property I want is that I'll call a diffusion Z regular if this domain is actually a core for its generator. Okay, so this is a certain technical condition on Markov process. And so here's our result which is let's suppose that there's some, you know, technical but not so important conditions on the domain and the two diffusions. Now importantly suppose that the solution Z to this, so this is a classic differential equation, is a regular fallar diffusion with the specified generator and also any F in this core satisfies this slightly horrific looking identity between integrals. Then in this case, Z is going to be an intertwining of X and Y for Gibbs initial conditions. Okay, so this is somewhat technical result and it's going to generalize the result of Paul and Shkolnikov in the case where all diffusions were driven by Brownian motions. And so I want to unpack this condition a little bit more. So for now just imagine it's some big inequality between integrals. So it's actually implied by three conditions which I think are a bit more interpretable. The first is that the generator should be intertwined by this Gibbs restriction lambda. So you should think of this first condition as some sort of differential version of this pitman intertwining. Then there are two conditions which are related to the fact that Z is a process with reflection. And so in particular it's reflection off a domain which depends on the top level. So this corresponds to the fact that in the Gelfand-Seven polytope if I fix the top level the domain of the lower levels actually changes. And so there should be some compatibility between the drift and diffusion terms and the way that the geometry of this shifting domain changes. And so for certain reasons these are the two conditions that you end up wanting to check. And in fact in our cases in most cases both sides are just equal to zero. But somehow this expresses some compatibility between the geometry of the space and the actual diffusions that you're running. So I think basically the first condition is interpretable and these are a bit less interpretable. But I will say that in the case where the driving terms are simply Brownian motions they reduce to just the conditions that were considered by Paul and Schoenberg. And so I want to talk a little bit about just very briefly about how to prove this result. So first the only thing that you need to prove is actually, or the key point that you need to prove is that the process Z is going to preserve Gibbs measures. And in fact what you need to show there is that for any function somehow this identity holds. Which means that if you evolve the process and apply the function that's the same thing as just evolving the top level of the process and finding the expectation of the function. So this first corresponds to just evolving the top level and finding the expectation of the function and the second corresponds to evolving the whole process and finding the expectation of the function. And if you actually take the infinitesimal version of this identity what you'll see is that you want to prove this somewhat easier identity. So it's some sort of relation between the generators of y and the generator of z. And so if you unpack this you'll see naturally this intertwining condition between the generators of x and y and this link function. And also you'll see that if you try to prove this by integration by parts what you'll obtain is first of all this thing and if you unpack this thing you'll see that you obtain these geometric compatibility terms. And so that's the reason why I say that these are somehow a relation between the domain of the diffusions and the actual driving terms of the diffusions. So I think I did quite a bit early but just to sum up basically what we did in this talk is first to relate the static structure of eigenvalue densities to the branching that comes from some special functions from representation theories and the multivariate Bessel and Hecht-Nampen functions. And that applied at beta equals one and two and somehow provides some continuation in beta of these matrix models. And the second part of the talk we talked about at beta equals two is the dynamical construction that reduces at a fixed time to these eigenvalue densities. So that's it. Questions? Yes, so one thing you can do is take a complex Brownian motion in the space of n by n Hermitian matrices. And then you can look at, for example, the eigenvalues and then the eigenvalues of the n by one times n by one principle sub-matrix. And so it was shown that that's Markov. But let's say you take the eigenvalues, the eigenvalues of the n by one by n by one matrix and the eigenvalues of the n by two by n by two matrix, then for some reason that's not Markov. And in fact, if you take any k principle sub-matrices where k is greater than two, then it's also not Markov. I think the answer is yes. However, I think the stochastic calculus gets... So the dynamics would have some, would be continuum time and continuum space and I think the stochastic calculus can get kind of complicated. So for Dyson Brownian motion there's beta deformation of it which was in this paper of Gorin and Shkolnikov. And there somehow the processes are very different than in the beta equals two case. For example, each particle will not have local interactions. Instead, the particle will depend on all particles on that level and the level below it. So these processes get a bit more complicated. I want to do that, but I have not done it. Yeah, there are some actually determinational formula for the joint distribution of the right edge. Other questions? If not, let's thank you again.