BetaEnsemblesSampling

Introduction

This Julia package provides methods to sample iid (independent and identically distributed) eigenvalues from the classical $\beta$ (beta) ensembles of random matrix theory (RMT). The parameter $\beta > 0$ plays the role of inverse temperature. So far the following multivariate eigenvalue distributions on $N$ ordered tuples $(\lambda_1, \dots, \lambda_N)$ are supported:

• Gaussian Beta Ensemble with probability distribution on $-\infty < \lambda_1 < \dots < \lambda_N < \infty$

$$P(\lambda_1, \dots, \lambda_N)d \lambda_1 \dots d \lambda_N \propto \prod_{1 \leq i < j \leq N} |\lambda_i - \lambda_j|^\beta \prod_{1 \leq i \leq N} e^{-\frac{(\lambda_i-\mu)^2}{2 \sigma^2}} d \lambda_i$$

with $\mu$ real and $\sigma, \beta &gt; 0.$

• Laguerre Beta Ensemble with probability distribution on $0 &lt; \lambda_1 &lt; \dots &lt; \lambda_N &lt; \infty$

$$P(\lambda_1, \dots, \lambda_N)d \lambda_1 \dots d \lambda_N \propto \prod_{1 \leq i < j \leq N} |\lambda_i - \lambda_j|^\beta \prod_{1 \leq i \leq N} \lambda_i^{\alpha-1} e^{-\lambda_i/\theta} d \lambda_i$$

with $\alpha, \theta, \beta &gt; 0.$

• Jacobi Beta Ensemble with probability distribution on $0 &lt; \lambda_1 &lt; \dots &lt; \lambda_N &lt; 1$

$$P(\lambda_1, \dots, \lambda_N)d \lambda_1 \dots d \lambda_N \propto \prod_{1 \leq i < j \leq N} |\lambda_i - \lambda_j|^\beta \prod_{1 \leq i \leq N} \lambda_i^{a-1} (1-\lambda_i)^{b-1} d \lambda_i$$

with $a, b, \beta &gt; 0.$

For generic $\beta$ these distributions are abbreviated $G \beta E$, $L \beta E$, and $J \beta E$ respectively. For $\beta = 1, 2, 4$ they are known as $G/L/JOE$, $G/L/JUE$, $G/L/JSE$, where O stands for orthogonal, U for unitary, and S for symplectic.

Usage

To sample num_samples = 10000 iid samples from the Laguerre Beta Ensemble above for matrix size N = 1000, and take only the smallest ordered num_taken = 100 eigenvalues, for beta = 4, alpha = 2.5, theta = 1 (from the LSE ensemble), you would do:

Lambda = LbetaE_eval_samples(N, num_taken, alpha, theta, beta, num_samples)

or

Lambda = LbetaE_eval_samples(1000, 100, 2.5, 1.0, 4.0, 10000)

Each of the num_samples rows of Lambda is an iid sample containing the first 100 eigenvalues.

Idea

While other packages providing similar functionality exist (see below), the idea is to provide a package that is:

• focused solely on sampling for researchers who want to experiment with classical random matrix models;
• fast;
• easy to use;
• easy to read;
• easy to modify for users unfamiliar with Julia;
• useful for batch sampling for situations in which taking a large number of iid samples (e.g. to compute expectations) might be needed.

Method

The sampling is done using the tridiagonal method of Dumitriu--Edelman and Killip--Nenciu, following the paper by Gautier--Bardenet--Valko (namely the conventions and notation of Thm 2.1, 2.2, and 2.3 of the latter). See the references for links.

Dependencies

It uses LinearAlgebra, Random, and Distributions.

Other similar packages

The Python package DPPy was originally implemented by Gautier--Bardenet--Valko as a companion to their paper referenced below.

The RandomMatrices.jl implements some of the same functionality as well.

References

Relevant references for the algorithms used

• Dumitriu, Edelman, Matrix models for beta ensembles, arXiv link
• Gautier, Bardenet, Valko, Fast sampling from beta ensembles, arXiv link
• Killip, Nenciu, Matrix models for circular ensembles, arXiv link

Relevant references on random matrix theory

• Anderson, Guionnet, Zeitouni, An introduction to random matrices, link to PDF
• Edelman, Eigenvalues and condition numbers of random matrices, PhD thesis, link to PDF
• Livan, Novaes, Vivo, An introduction to random matrices - theory and practice, arXiv link