Sampling Pfaffian point processes and the symplectic Arnoldi method
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| Format: | Preprint |
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2026
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| _version_ | 1866911640005378048 |
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| author | Edelman, Alan Jeong, Sungwoo Schaub, Simeon |
| author_facet | Edelman, Alan Jeong, Sungwoo Schaub, Simeon |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_01202 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sampling Pfaffian point processes and the symplectic Arnoldi method Edelman, Alan Jeong, Sungwoo Schaub, Simeon Numerical Analysis 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. |
| title | Sampling Pfaffian point processes and the symplectic Arnoldi method |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2605.01202 |