arXiv subjects

Hoang Dung Trinh

Publications and source records attributed to Hoang Dung Trinh.

7 recordsLinked to original sources

arXiv2026

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

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.

math-ph↗

arXiv2025

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

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.

math.PR↗

arXiv2023

Beta Jacobi ensembles and associated Jacobi polynomials, II

In a high temperature regime, it was shown in Trinh--Trinh (\emph{J.\ Stat.\ Phys.}\ \textbf{185}(1), Paper No.\ 4, 15 (2021)) that the empirical distribution of beta Jacobi ensembles converges to a limiting probability measure which is related to Model III of associated Jacobi polynomials. In this paper, we establish Gaussian fluctuations around the limit whose statement involves orthogonal polynomials. For the proofs, we refine a moment method at the process level which has been used to deal with Gaussian beta ensembles and beta Laguerre ensembles.

math.PR↗

arXiv2023

Limit theorems for moment processes of beta Dyson's Brownian motions and beta Laguerre processes

In the regime where the parameter beta is proportional to the reciprocal of the system size, it is known that the empirical distribution of Gaussian beta ensembles (resp.\ beta Laguerre ensembles) converges to a probability measure of associated Hermite polynomials (resp.\ associated Laguerre polynomials). Gaussian fluctuations around the limit have been known as well. This paper aims to study a dynamical version of those results. More precisely, we study beta Dyson's Brownian motions and beta Laguerre processes and establish LLNs and CLTs for their moment processes in the same regime.

math.PR↗

arXiv2021

Beta Jacobi ensembles and associated Jacobi polynomials

Beta ensembles on the real line with three classical weights (Gaussian, Laguerre and Jacobi) are now realized as the eigenvalues of certain tridiagonal random matrices. The paper deals with beta Jacobi ensembles, the type with the Jacobi weight. Making use of the random matrix model, we show that in the regime where $βN \to const \in [0, \infty)$, with $N$ the system size, the empirical distribution of the eigenvalues converges weakly to a limiting measure which belongs to a new class of probability measures of associated Jacobi polynomials. This is analogous to the existing results for the other two classical weights. We also study the limiting behavior of the empirical measure process of beta Jacobi processes in the same regime and obtain a dynamic version of the above.

math.PR↗

arXiv2021

Beta Laguerre processes in a high temperature regime

Beta Laguerre processes which are generalizations of the eigenvalue process of Wishart/Laguerre processes can be defined as the square of radial Dunkl processes of type B. In this paper, we study the limiting behavior of their empirical measure processes. By the moment method, we show the convergence to a limit in a high temperature regime, a regime where $βN \to const \in (0, \infty)$, where $β$ is the inverse temperature parameter and $N$ is the system size. This is a dynamic version of a recent result on the convergence of the empirical measures of beta Laguerre ensembles in the same regime.

math.PR↗

arXiv2019

Beta Laguerre ensembles in global regime

Beta Laguerre ensembles which are generalizations of Wishart ensembles and Laguerre ensembles can be realized as eigenvalues of certain random tridiagonal matrices. Analogous to the Wishart ($β=1$) case and the Laguerre ($β= 2$) case, for fixed $β$, it is known that the empirical distribution of the eigenvalues of these ensembles converges weakly to Marchenko--Pastur distributions, almost surely. The paper restudies the limiting behavior of the empirical distribution but in regimes where the parameter $β$ is allowed to vary as a function of the matrix size $N$. We show that the above Marchenko--Pastur law holds as long as $βN \to \infty$. When $βN \to 2c \in (0, \infty)$, the limit is related to associated Laguerre orthogonal polynomials. Gaussian fluctuations around the limit are also studied.

math.PR↗