The intertwining property for $β$-Laguerre processes and integral operators for Jack polynomials
Yosuke Kawamato, Genki Shibukawa
TL;DR
This work develops new fixed-parameter intertwining relations for $\beta$-Laguerre processes by introducing a Markov kernel depending on $\theta=\beta/2$ and $\alpha>-1$, and proves that Jack polynomials are eigenfunctions of these kernels. The authors adapt a Ramanan–Shkolnikov framework to obtain generator and semigroup intertwinings, with a crucial eigenrelation $\Lambda_{\theta,\alpha,N}^{N}P_\lambda^\theta=c(\lambda,N,\theta;\alpha)P_\lambda^\theta$, enabling explicit action on multivariate Laguerre polynomials and related hypergeometric functions. A probabilistic interpretation for classical $\theta$ values ($\theta\in\{1/2,1,2\}$) is given via radial parts of invariant random matrices, linking the kernels to truncated Haar matrix ensembles and Laguerre-type distributions. The results extend intertwining theory for Laguerre processes, provide new integral formulas, and deepen connections between stochastic processes, symmetric function theory, and random matrix theory. These insights may impact the study of spectral processes and representations of Jack polynomials in high-dimensional stochastic systems.
Abstract
The aim of this paper is to study intertwining relations for Laguerre process with inverse temperature $β\ge 1$ and parameter $α>-1$. We introduce a Markov kernel that depends on both $β$ and $ α$, and establish new intertwining relations for the $β$-Laguerre processes using this kernel. A key observation is that Jack symmetric polynomials are eigenfunctions of our Markov kernel, which allows us to apply a method established by Ramanan and Shkolnikov. Additionally, as a by-product, we derive an integral formula for multivariate Laguerre polynomials and multivariate hypergeometric functions associated with Jack polynomials.