Non-linear PDEs for gap probabilities in random matrices and KP theory

2011/04/30 by Mark Adler, M. Adler, Mattia Cafasso +3 · 1 citation
Decision Sciences · Mathematics · Physics and Astronomy · #Applied mathematics #Eigenvalues and eigenvectors #Mathematics #Physics #Probability and Risk Models #Quantum mechanics #Random Matrices and Applications #Random matrix #Statistical physics #advanced mathematical theories #math-ph #math.MP #math.PR #nlin.SI

paper · pdf · doi:10.1016/j.physd.2012.08.016

published as Physica D, 241, 2012, 2265-2284 · Minor revision: accepted for publication on Physica D

openalex publication_date 2012/08/27 · arxiv created 2012/10/25 · arxiv updated 2013/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Airy and Pearcey-like kernels and generalizations arising in random matrix theory are expressed as double integrals of ratios of exponentials, possibly multiplied with a rational function. In this work it is shown that such kernels are intimately related to wave functions for polynomial (Gel'fand-Dickey reductions) or rational reductions of the KP-hierarchy; their Fredholm determinant also satisfies linear PDEs (Virasoro constraints), yielding, in a systematic way, non-linear PDEs for the Fredholm determinant of such kernels. Examples include Fredholm determinants giving the gap probability of some infinite-dimensional diffusions, like the Airy process, with or without outliers, and the Pearcey process, with or without inliers.

Citations

• Universal singularity at the closure of a gap in a random matrix theory
• Differential Equations for Dyson Processes
• From the Pearcey to the Airy Process.
• The vector k-constrained KP hierarchy and Sato's Grassmannian
• Riemann–Hilbert approach to multi-time processes: The Airy and the Pearcey cases
• DIFFERENTIAL EQUATIONS FOR QUANTUM CORRELATION FUNCTIONS
• PDEs for the joint distributions of the Dyson, Airy and Sine processes
• Extension of level-spacing universality

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