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Classical beta ensembles and related eigenvalues processes at high temperature and the Markov--Krein transform

Fumihiko Nakano, Hoang Dung Trinh, Khanh Duy Trinh

TL;DR

The paper investigates the high-temperature limits (β = 2c/N) of the classical beta ensembles—Gaussian, Laguerre, and Jacobi—through the Markov--Krein transform (MKR) framework, showing that the limiting one-point measures μ_c are the MKR inverses of the standard Gaussian, Gamma, and Beta distributions, respectively. At the process level, the authors identify the limiting measure-valued processes μ_t as inverse MKRs of one-dimensional diffusions, notably L(ξ + b_t) for the Gaussian case, with analogous diffusion representations for Laguerre and Jacobi settings, established via both direct PDE arguments and formal series methods. They develop moment relations in MKR and introduce the concept of c-convolution to interpolate between classical and free convolution, providing a unifying mechanism for the high-temperature behavior of eigenvalue processes. The work also analyzes mean-field limits and local scaling regimes (Gaussian and Laguerre-local, Jacobi-local) for beta Dyson’s Brownian motions and related Laguerre/Jacobi processes, connecting spectral limits to MKR-driven dynamics and highlighting the role of Stieltjes transforms and moment recursions. Overall, the paper builds a coherent MKR-based framework for understanding how classical random matrix ensembles transition to transform-based limits under large N and shrinking β, with explicit stochastic representations of the limiting objects and convergence results useful for mean-field theory and spectral analytics.

Abstract

The aim of this paper is to identify the limit in a high temperature regime of classical beta ensembles on the real line and related eigenvalue processes by using the Markov--Krein transform. We show that the limiting measure of Gaussian beta ensembles (resp.\ beta Laguerre ensembles and beta Jacobi ensembles) is the inverse Markov--Krein transform of the Gaussian distribution (resp.\ the gamma distribution and the beta distribution). At the process level, we show that the limiting probability measure-valued process is the inverse Markov--Krein transform of a certain 1d stochastic process.

Classical beta ensembles and related eigenvalues processes at high temperature and the Markov--Krein transform

TL;DR

The paper investigates the high-temperature limits (β = 2c/N) of the classical beta ensembles—Gaussian, Laguerre, and Jacobi—through the Markov--Krein transform (MKR) framework, showing that the limiting one-point measures μ_c are the MKR inverses of the standard Gaussian, Gamma, and Beta distributions, respectively. At the process level, the authors identify the limiting measure-valued processes μ_t as inverse MKRs of one-dimensional diffusions, notably L(ξ + b_t) for the Gaussian case, with analogous diffusion representations for Laguerre and Jacobi settings, established via both direct PDE arguments and formal series methods. They develop moment relations in MKR and introduce the concept of c-convolution to interpolate between classical and free convolution, providing a unifying mechanism for the high-temperature behavior of eigenvalue processes. The work also analyzes mean-field limits and local scaling regimes (Gaussian and Laguerre-local, Jacobi-local) for beta Dyson’s Brownian motions and related Laguerre/Jacobi processes, connecting spectral limits to MKR-driven dynamics and highlighting the role of Stieltjes transforms and moment recursions. Overall, the paper builds a coherent MKR-based framework for understanding how classical random matrix ensembles transition to transform-based limits under large N and shrinking β, with explicit stochastic representations of the limiting objects and convergence results useful for mean-field theory and spectral analytics.

Abstract

The aim of this paper is to identify the limit in a high temperature regime of classical beta ensembles on the real line and related eigenvalue processes by using the Markov--Krein transform. We show that the limiting measure of Gaussian beta ensembles (resp.\ beta Laguerre ensembles and beta Jacobi ensembles) is the inverse Markov--Krein transform of the Gaussian distribution (resp.\ the gamma distribution and the beta distribution). At the process level, we show that the limiting probability measure-valued process is the inverse Markov--Krein transform of a certain 1d stochastic process.

Paper Structure

This paper contains 19 sections, 21 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. Classical beta ensembles at high temperature
  3. Gaussian beta ensembles
  4. Beta Laguerre ensembles
  5. Beta Jacobi ensembles
  6. Moment relations in the MKR
  7. Beta Dyson's Brownian motions
  8. Identifying the limiting process
  9. A direct approach
  10. A formal series approach
  11. Beta Laguerre processes
  12. Mean-field limit for beta Laguerre processes
  13. An attempt on a direct method
  14. A formal series approach
  15. Local scaling regime
  16. ...and 4 more sections

Key Result

Theorem 1.1

Denote by $\mu_c^{(G)}$, $\mu_c^{(L)}$ and $\mu_c^{(J)}$ the limiting measure in the high temperature regime $\beta = 2c/N$ of the empirical distribution of Gaussian beta ensembles, beta Laguerre ensembles and beta Jacobi ensembles joint-density, respectively. Here $c\in (0, \infty)$ is given. Then

Theorems & Definitions (40)

  • Theorem 1.1
  • Lemma 2.1
  • Remark 2.2
  • proof
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.4
  • ...and 30 more