Sampling Pfaffian point processes and the symplectic Arnoldi method

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 (${\hat{I}}^2=1,4$). 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 ${\hat{I}}^2=1,4$ 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 ${\hat{I}}^2=1,4$ Airy point processes and Tracy-Widom distributions.