# On a gateway between the Laguerre process and dynamics on partitions

## TL;DR

This paper establishes exact relations between the Laguerre process eigenvalues and dynamics on partitions, connecting matrix eigenvalues with combinatorial structures through generalized diffusion results.

## Contribution

It generalizes recent one-dimensional diffusion results to multidimensional settings, linking Laguerre eigenvalues with partition dynamics and ensembles.

## Key findings

- Derived exact relations between Laguerre eigenvalues and partition dynamics.
- Connected Laguerre and Meixner ensembles explicitly.
- Explored links with Young bouquet and z-measures on partitions.

## Abstract

Probability measures and stochastic dynamics on matrices and on partitions are related by standard, albeit technical, discrete to continuous scaling limits. In this paper we provide exact relations, that go in both directions, between the eigenvalues of the Laguerre process and certain distinguished dynamics on partitions. This is done by generalizing to the multidimensional setting recent results of Miclo and Patie on linear one-dimensional diffusions and birth and death chains. As a corollary, we obtain an exact relation between the Laguerre and Meixner ensembles. Finally, we explain the deep connections with the Young bouquet and the z-measures on partitions.