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The intertwining property for Laguerre processes with a fixed parameter

Alexander I. Bufetov, Yosuke Kawamoto

Abstract

We investigate the intertwining of Laguerre processes of parameter $α$ in different dimensions. We introduce a Feller kernel that depends on $α$ and intertwines the $α$-Laguerre process in $N+1$ dimensions and that in $N$ dimensions. When $α$ is a non-negative integer, the new kernel is interpreted in terms of the conditional distribution of the squared singular values: if the singular values of a unitarily invariant random matrix of order $(N+α+1) \times (N+1)$ are fixed, then the those of its $(N+α) \times N $ truncation matrix are given by the new kernel.

The intertwining property for Laguerre processes with a fixed parameter

Abstract

We investigate the intertwining of Laguerre processes of parameter in different dimensions. We introduce a Feller kernel that depends on and intertwines the -Laguerre process in dimensions and that in dimensions. When is a non-negative integer, the new kernel is interpreted in terms of the conditional distribution of the squared singular values: if the singular values of a unitarily invariant random matrix of order are fixed, then the those of its truncation matrix are given by the new kernel.
Paper Structure (11 sections, 16 theorems, 72 equations)

This paper contains 11 sections, 16 theorems, 72 equations.

Table of Contents

  1. Introduction and main results
  2. Intertwining relations for the Laguerre processes
  3. Interpretation of Markov kernels in terms of radial parts of random matrices
  4. Organisation of this paper
  5. Feller kernels
  6. Kernels and semigroups
  7. Feller property of the Markov kernels $\Lambda_{\alpha, N}^{N+1}$
  8. Intertwining for the Laguerre processes
  9. The Laguerre processes
  10. Proof of Theorem \ref{['t:21']}
  11. Proof of Theorem \ref{['t:12']}

Key Result

Theorem 1.1

Assume that $\alpha >-1$. Then, for any $N\in\mathbb{N}$, $f\in C_{\infty} (W_{\ge}^{N} )$, and $t \ge 0$, we have the intertwining relation

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • proof
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 19 more