arXiv subjects

Khanh Duy Trinh

Publications and source records attributed to Khanh Duy Trinh.

16 recordsLinked to original sources

arXiv2026

Beta Ensembles in the Freezing Regime and Finite Free Convolutions

In the freezing regime where the system size N is fixed and the inverse temperature beta tends to infinity, the eigenvalues of Gaussian beta ensembles converge to zeros of the Nth Hermite polynomial. That law of large numbers has been proved by analyzing the joint density or reading off the random matrix model. This paper studies its dynamical version of this phenomenon. We show that in the freezing regime the eigenvalue processes called beta Dyson Brownian motions converge to deterministic limiting processes which can be written as the finite free convolution of the initial data and the zeros of Hermite polynomials. This result is a counterpart of those in the random matrix regime where N tends to infinity with fixed beta, as well as to the high temperature regime where N tends to infinity while beta N remains bounded. We also establish Gaussian fluctuations around the limit and deal with the Laguerre case.

math.PR↗

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↗

arXiv2024

Normal approximation for statistics of randomly weighted complexes

We prove normal approximation bounds for statistics of randomly weighted (simplicial) complexes. In particular, we consider the complete $d$-dimensional complex on $n$ vertices with $d$-simplices equipped with i.i.d. weights. Our normal approximation bounds are quantified in terms of stabilization of difference operators, i.e., the effect on the statistic under addition/deletion of simplices. Our proof is based on Chatterjee's normal approximation bound and is a higher-dimensional analogue of the work of Cao on sparse Erdős-Rényi random graphs but our bounds are more in the spirit of `quantitative two-scale stabilization' bounds by Lachièze-Rey, Peccati, and Yang. As applications, we prove a CLT for nearest face-weights in randomly weighted $d$-complexes and give a normal approximation bound for local statistics of random $d$-complexes.

math.PR↗

arXiv2023

Central limit theorem for linear eigenvalue statistics of the adjacency matrices of random simplicial complexes

We study the adjacency matrix of the Linial-Meshulam complex model, which is a higher-dimensional generalization of the Erdős-Rényi graph model. Recently, Knowles and Rosenthal proved that the empirical spectral distribution of the adjacency matrix is asymptotically given by Wigner's semicircle law in a diluted regime. In this article, we prove a central limit theorem for the linear eigenvalue statistics for test functions of polynomial growth that is of class $C^{2}$ on a closed interval. The proof is based on higher-dimensional combinatorial enumerations and concentration properties of random symmetric matrices. Furthermore, when the test function is a polynomial function, we obtain the explicit formula for the variance of the limiting Gaussian distribution.

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↗

arXiv2021

Random connection models in the thermodynamic regime: central limit theorems for add-one cost stabilizing functionals

The paper deals with a random connection model, a random graph whose vertices are given by a homogeneous Poisson point process on $\mathbb{R}^d$, and edges are independently drawn with probability depending on the locations of the two end points. We establish central limit theorems (CLT) for general functionals on this graph under minimal assumptions that are a combination of the weak stabilization for the-one cost and a $(2+δ)$-moment condition. As a consequence, CLTs for isomorphic subgraph counts, isomorphic component counts, the number of connected components are then derived. In addition, CLTs for Betti numbers and the size of biggest component are also proved for the first time.

math.PR↗

arXiv2020

Poisson statistics for beta ensembles on the real line at high temperature

This paper studies beta ensembles on the real line in a high temperature regime, that is, the regime where $βN \to const \in (0, \infty)$, with $N$ the system size and $β$ the inverse temperature. In this regime, the convergence to the equilibrium measure is a consequence of a recent result on large deviation principle by Liu and Wu (Stochastic Processes and their Applications (2019)). This paper focuses on the local behavior and shows that the local statistics around any fixed reference energy converges weakly to a homogeneous Poisson point process.

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↗

arXiv2019

On persistent homology of random Čech complexes

The paper studies the relation between critical simplices and persistence diagrams of the Čech filtration. We show that adding a critical $k$-simplex into the filtration corresponds either to a point in the $k$th persistence diagram or a point in the $(k-1)$st persistence diagram. Consequently, the number of points in persistence diagrams can be expressed in terms of the number of critical simplices. As an application, we establish some convergence results related to persistence diagrams of the Čech filtrations built over binomial point processes.

math.PR↗

arXiv2018

Strong law of large numbers for Betti numbers in the thermodynamic regime

We establish the strong law of large numbers for Betti numbers of random Čech complexes built on $\mathbb R^N$-valued binomial point processes and related Poisson point processes in the thermodynamic regime. Here we consider both the case where the underlying distribution of the point processes is absolutely continuous with respect to the Lebesgue measure on $\mathbb R^N$ and the case where it is supported on a $C^1$ compact manifold of dimension strictly less than $N$. The strong law is proved under very mild assumption which only requires that the common probability density function belongs to $L^p$ spaces, for all $1\leq p < \infty$.

math.PR↗

arXiv2018

On central limit theorems in stochastic geometry

We establish central limit theorems for general functionals on binomial point processes and their Poissonized version. As an application, a central limit theorem for Betti numbers of random geometric complexes in the thermodynamic regime is derived.

math.PR↗

arXiv2017

Global spectrum fluctuations for Gaussian beta ensembles: a martingale approach

The paper describes the global limiting behavior of Gaussian beta ensembles where the parameter $β$ is allowed to vary with the matrix size $n$. In particular, we show that as $n \to \infty$ with $nβ\to \infty$, the empirical distribution converges weakly to the semicircle distribution, almost surely. The Gaussian fluctuation around the limit is then derived by a martingale approach.

math.PR↗