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The spectral measures of random Jacobi matrices related to beta ensembles at high temperature and Dirichlet processes

Fumihiko Nakano, Hoang Dung Trinh, Khanh Duy Trinh

TL;DR

This work analyzes spectral measures of random Jacobi matrices associated with Gaussian, Laguerre, and Jacobi beta ensembles in the high-temperature regime $βN \to 2c$. It proves that the spectral measures converge in distribution to Dirichlet processes $\mathrm{DP}(\rho_c, c)$ (or their appropriate analogs), with base distributions $\rho_c$ determined by associated polynomials. The semi-infinite Jacobi limit $H_c$ also exhibits a Dirichlet-process spectral law, yielding explicit examples of random Jacobi matrices whose spectral measures are known objects. The results combine the Dirichlet process construction, Markov–Krein transforms, and finite-dimensional tridiagonal models to provide a unified framework for understanding spectral measures in high-temperature beta ensembles. This offers precise probabilistic representations of limiting spectral data and connects random matrix theory with Dirichlet-process methods.

Abstract

In a high temperature regime where $βN \to 2c$, the empirical distribution of the eigenvalues of Gaussian beta ensembles, beta Laguerre ensembles and beta Jacobi ensembles converges to a limiting measure which is related to associated Hermite polynomials, associated Laguerre polynomials and associated Jacobi polynomials, respectively. Here $β$ is the inverse temperature parameter, $N$ is the system size and $c>0$ is a given constant. This paper studies the spectral measure of the random tridiagonal matrix model of the three classical beta ensembles. We show that in the high temperature regime, the spectral measure converges in distribution to a Dirichlet process with base distribution being the limiting distribution, and scaling parameter $c$. Consequently, the spectral measure of a related semi-infinite Jacobi matrix coincides with that Dirichlet process, which provides examples of random Jacobi matrices with explicit spectral measures.

The spectral measures of random Jacobi matrices related to beta ensembles at high temperature and Dirichlet processes

TL;DR

This work analyzes spectral measures of random Jacobi matrices associated with Gaussian, Laguerre, and Jacobi beta ensembles in the high-temperature regime . It proves that the spectral measures converge in distribution to Dirichlet processes (or their appropriate analogs), with base distributions determined by associated polynomials. The semi-infinite Jacobi limit also exhibits a Dirichlet-process spectral law, yielding explicit examples of random Jacobi matrices whose spectral measures are known objects. The results combine the Dirichlet process construction, Markov–Krein transforms, and finite-dimensional tridiagonal models to provide a unified framework for understanding spectral measures in high-temperature beta ensembles. This offers precise probabilistic representations of limiting spectral data and connects random matrix theory with Dirichlet-process methods.

Abstract

In a high temperature regime where , the empirical distribution of the eigenvalues of Gaussian beta ensembles, beta Laguerre ensembles and beta Jacobi ensembles converges to a limiting measure which is related to associated Hermite polynomials, associated Laguerre polynomials and associated Jacobi polynomials, respectively. Here is the inverse temperature parameter, is the system size and is a given constant. This paper studies the spectral measure of the random tridiagonal matrix model of the three classical beta ensembles. We show that in the high temperature regime, the spectral measure converges in distribution to a Dirichlet process with base distribution being the limiting distribution, and scaling parameter . Consequently, the spectral measure of a related semi-infinite Jacobi matrix coincides with that Dirichlet process, which provides examples of random Jacobi matrices with explicit spectral measures.
Paper Structure (9 sections, 9 theorems, 101 equations)

This paper contains 9 sections, 9 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Diriclet processes and the Markov--Krein transform
  3. Proof of Theorem \ref{['thm:intro-general']}
  4. Classical beta ensembles on the real line
  5. Gaussian beta ensembles
  6. Beta Laguerre ensembles
  7. Beta Jacobi ensembles
  8. Concluding remarks on beta ensembles on the real line
  9. Convergence of random probability measures on the real line

Key Result

Theorem 1.1

Let $c > 0$ be given. For each $N$, let $(w_1, \dots, w_N)$ have the symmetric Dirichlet distribution with parameter $c/N$ and be independent of random variables $\lambda_1, \dots, \lambda_N$. Assume that the empirical distribution $L_N = N^{-1} \sum_{i=1}^N \delta_{\lambda_i}$ converges weakly to a converges in distribution to a Dirichlet process with base distribution $\rho$ and scaling paramete

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Example 2.1
  • proof : Proof of the MKR \ref{['MKR-original']}
  • Example 3.1
  • proof : Proof of Theorem \ref{['thm:intro-general']}
  • Lemma 4.1
  • proof
  • Remark 4.2
  • Theorem 4.3
  • ...and 10 more