Sampling Pfaffian point processes and the symplectic Arnoldi method

TL;DR

This paper introduces an exact sampling algorithm for Pfaffian point processes using a skew-symmetric Cholesky factorization and a symplectic Arnoldi method for computing skew-orthogonal polynomials, enabling efficient sampling in random matrix theory.

Contribution

It presents a novel exact sampling algorithm for Pfaffian point processes and a symplectic Arnoldi method for constructing skew-orthogonal polynomials, advancing computational techniques in random matrix theory.

Findings

Efficient sampling of eigenvalues in orthogonal and symplectic ensembles.

Numerical examples demonstrate the effectiveness of the methods.

The approach applies to various models including Tracy-Widom distributions.

Abstract

We present an exact sampling algorithm for Pfaffian point processes based on a skew-symmetric analogue of the Cholesky factorization. This algorithm enables efficient sampling of a wide range of statistics arising in random matrix theory and combinatorics. For instance, we can sample eigenvalues of the orthogonal and symplectic ensembles ($\beta = 1,4$). In addition, we introduce a symplectic Arnoldi method for computing skew-orthogonal polynomials associated with a general weight function. This method can be used to efficiently construct the $2 \times 2$ matrix valued skew-symmetric kernels that arise in $\beta = 1,4$ polynomial ensembles. We illustrate our approach with several numerical examples and experiments, including the symmetric corner growth model, the finite-$N$ Gaussian (Hermite) orthogonal and symplectic ensembles, and the $\beta = 1,4$ Airy point processes and Tracy-Widom distributions.