[Paper Review] Gaussian beta ensembles at high temperature: eigenvalue fluctuations and bulk statistics

Trinh Khanh Duy, Fumihiko Nakano|arXiv (Cornell University)|Nov 29, 2016

Random Matrices and Applications14 references3 citations

TL;DR

This paper investigates Gaussian beta ensembles in the high-temperature limit where $\beta n = \text{const}$ as $n \to \infty$. It establishes Gaussian fluctuations for linear eigenvalue statistics and proves Poisson convergence of bulk eigenvalue statistics, offering an alternative derivation with explicit intensity measures, extending semi-circle law behavior to low-temperature regimes.

ABSTRACT

We study the limiting behavior of Gaussian beta ensembles in the regime where $βn = const$ as $n o \infty$. The results are (1) Gaussian fluctuations for linear statistics of the eigenvalues, and (2) Poisson convergence of the bulk statistics. (2) is an alternative proof of the result by F.~Benaych-Georges and S.~Péché (2015) with the explicit form of the intensity measure.

Motivation & Objective

• To analyze the limiting spectral behavior of Gaussian beta ensembles when $n \to \infty$ and $\beta \to 0$ with $n\beta = \text{const}$, corresponding to a high-temperature regime.
• To establish central limit theorem-type results for linear eigenvalue statistics in this regime.
• To investigate the bulk scaling limit of eigenvalues and determine its limiting point process.
• To provide an alternative proof of Poisson convergence for bulk statistics, previously shown by Benaych-Georges and Péché (2015), with explicit intensity measure.
• To clarify the relationship between empirical and spectral measures in the joint limit, showing convergence to a non-random limit in probability

Proposed method

• Uses the matrix model of Dumitriu and Edelman for Gaussian beta ensembles, represented as symmetric tridiagonal Jacobi matrices with $\mathcal{N}(0,1)$ diagonal and $\tilde{\chi}_{(n-k)\beta}$ subdiagonal entries.
• Analyzes the joint limit $n \to \infty$, $\beta \to 0$ with $n\beta = 2\alpha$ fixed, showing convergence of matrix entries to i.i.d. entries of an infinite Jacobi matrix $J_\alpha$.
• Applies the spectral measure theory for infinite Jacobi matrices, using the condition $\sum_{i=1}^\infty 1/b_i = \infty$ to ensure uniqueness of the spectral measure $\mu_\alpha$ almost surely.
• Establishes weak convergence of the empirical measure $L_{n,\beta}$ to $\bar{\mu}_\alpha$ in probability and of the spectral measure $\nu_{n,\beta}$ to $\mu_\alpha$ in distribution.
• Employs convergence determining classes in $C_b(\mathbb{R})$ and moment-based criteria to prove weak convergence of random measures, leveraging bounded convergence and moment convergence.
• Uses the fact that eigenvalue weights $q_j^2$ are Dirichlet-distributed with parameter $\beta/2$, independent of eigenvalues, to show $\mathbb{E}[L_{n,\beta}] = \mathbb{E}[\nu_{n,\beta}]$ and to analyze fluctuations

Experimental results

Research questions

• RQ1 What is the limiting behavior of linear eigenvalue statistics in Gaussian beta ensembles when $\beta n = \text{const}$ as $n \to \infty$?
• RQ2 How do the bulk eigenvalue statistics behave in the high-temperature regime ($\beta \to 0$) with $n\beta$ fixed?
• RQ3 Can the Poisson convergence of bulk statistics be re-derived with an explicit expression for the intensity measure?
• RQ4 How do the empirical and spectral measures of $T_{n,\beta}$ behave in the joint limit $n \to \infty$, $\beta \to 0$ with $n\beta = \text{const}$?
• RQ5 What is the relationship between the mean of the empirical measure and the mean of the spectral measure in this regime?

Key findings

• In the regime $n\beta = 2\alpha = \text{const}$, the empirical measure $L_{n,\beta}$ converges weakly to $\bar{\mu}_\alpha$ in probability, where $\bar{\mu}_\alpha$ is the mean of the spectral measure of the infinite Jacobi matrix $J_\alpha$.
• The spectral measure $\nu_{n,\beta}$ of $T_{n,\beta}$ converges in distribution to the spectral measure $\mu_\alpha$ of $J_\alpha$, which is almost surely unique due to the divergence of $\sum 1/b_i$.
• Linear eigenvalue statistics converge to a Gaussian distribution, establishing a central limit theorem for fluctuations in the high-temperature limit.
• The bulk statistics of eigenvalues converge to a Poisson point process with explicit intensity measure, providing a new proof of a result by Benaych-Georges and Péché (2015).
• The mean of the empirical measure and the mean of the spectral measure coincide: $\bar{L}_{n,\beta} = \bar{\nu}_{n,\beta}$, due to the Dirichlet nature of the eigenvalue weights.
• The convergence of $L_{n,\beta}$ to $\bar{\mu}_\alpha$ in probability is established via moment convergence and the bounded convergence theorem applied to the expectation of linear statistics.