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arXiv:2312.02685 (Published 2023-12-05)
On the differential equations of frozen Calogero-Moser-Sutherland particle models
Multivariate Bessel and Jacobi processes describe Calogero-Moser-Sutherland particle models. They depend on a parameter $k$ and are related to time-dependent classical random matrix models like Dysom Brownian motions, where $k$ has the interpretation of an inverse temperature. There are several stochastic limit theorems for $k\to\infty$ were the limits depend on the solutions of associated ODEs where these ODEs admit particular simple solutions which are connected with the zeros of the classical orthogonal polynomials. In this paper we show that these solutions attract all solutions. Moreover we present a connection between the solutions of these ODEs with associated inverse heat equations. These inverse heat equations are used to compute the expectations of some determinantal formulas for the Bessel and Jacobi processes.
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arXiv:2203.07797 (Published 2022-03-15)
Wigner- and Marchenko-Pastur-type limits for Jacobi processes
We study Jacobi processes $(X_{t})_{t\ge0}$ on the compact spaces $[-1,1]^N$ and on the noncompact spaces $[1,\infty[^N$ which are motivated by the Heckman-Opdam theory for the root systems of type BC and associated integrable particle systems. These processes depend on three positive parameters and degenerate in the freezing limit to solutions of deterministic dynamical systems. In the compact case, these models tend for $t\to\infty$ to the distributions of the $\beta$-Jacobi ensembles and, in the freezing case, to vectors consisting of ordered zeros of one-dimensional Jacobi polynomials. Representing these processes by stochastic differential equations, we derive almost sure analogues of Wigner's semicircle and Marchenko-Pastur limit laws for $N\to\infty$ for the empirical distributions of the $N$ particles on some local scale. We there allow for arbitrary initial conditions, which enter the limiting distributions via free convolutions These results generalize corresponding stationary limit results in the compact case for $\beta$-Jacobi ensembles and, in the deterministic case, for the empirical distributions of the ordered zeros of Jacobi polynomials by Dette and Studden. The results are also related to free limit theorems for multivariate Bessel processes, $\beta$-Hermite and $\beta$-Laguerre ensembles, and the asymptotic empirical distributions of the zeros of Hermite and Laguerre polynomials for $N\to\infty$.
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arXiv:1609.03764 (Published 2016-09-13)
Intertwinings for general $β$ Laguerre and Jacobi processes
We show that for $\beta \ge 1$ the semigroups of $\beta$ Laguerre and $\beta$ Jacobi processes of different dimensions are intertwined in analogy to a similar result for $\beta$ Dyson Brownian motion recently obtained by Ramanan and Shkolnikov. These intertwining relations generalize to arbitrary $\beta \ge 1$ the ones obtained for $\beta=2$ by the author, O'Connell and Warren between $h$-transformed Karlin-McGregor semigroups. Moreover they form the key first step towards constructing a multilevel process in a Gelfand Tsetlin pattern. Finally as a by product we obtain a relation between general $\beta$ Jacobi ensembles of different dimensions.
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arXiv:1201.3490 (Published 2012-01-17)
Central limit theorems for hyperbolic spaces and Jacobi processes on $[0,\infty[$
We present a unified approach to a couple of central limit theorems for radial random walks on hyperbolic spaces and time-homogeneous Markov chains on the positive half line whose transition probabilities are defined in terms of the Jacobi convolutions. The proofs of all results are based on limit results for the associated Jacobi functions. In particular, we consider the cases where the first parameter (i.e., the dimension of the hyperbolic space) tends to infinity as well as the cases $\phi_{i\rho-\lambda}^{(\alpha,\beta)}(t)$ for small $\lambda$, and $\phi_{i\rho-n\lambda}^{(\alpha,\beta)}(t/n)$ for $n\to\infty$. The proofs of all these limit results are based on the known Laplace integral representation for Jacobi functions. Parts of the limit results for Jacobi functions and of the CLTs are known, other improve known ones, and other are completely new.