Sampling Pfaffian point processes and the symplectic Arnoldi method
This work provides a practical tool for sampling from Pfaffian point processes, which is important for researchers in random matrix theory and combinatorics, though it is an incremental algorithmic contribution.
The authors present an exact sampling algorithm for Pfaffian point processes, enabling efficient sampling of eigenvalues for orthogonal and symplectic ensembles (β=1,4), and introduce a symplectic Arnoldi method for computing skew-orthogonal polynomials. Numerical examples demonstrate the approach on models like the symmetric corner growth model and Tracy-Widom distributions.
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 ($β= 1,4$). 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 $β= 1,4$ 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 $β= 1,4$ Airy point processes and Tracy-Widom distributions.