Nonlinear PDEs for gap probabilities in random matrices and KP theory
arXiv:1104.4268 · doi:10.1016/j.physd.2012.08.016
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.
Minor revision: accepted for publication on Physica D
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Cited by in corpus (12)
- Multicritical edge statistics for the momenta of fermions in non-harmonic traps
- Large Complex Correlated Wishart Matrices: The Pearcey Kernel and Expansion at the Hard Edge
- Fredholm determinant solutions of the Painlevé II hierarchy and gap probabilities of determinantal point processes
- Higher Order Analogues of Tracy-Widom Distributions via the Lax Method
- A Survey on the Eigenvalues Local Behavior of Large Complex Correlated Wishart Matrices
- Darboux transformations and random point processes
- KP integrability of triple Hodge integrals. II. Generalized Kontsevich matrix model
- Differential equations for the KPZ and periodic KPZ fixed points
- Gap probability for the hard edge Pearcey process
- On higher Brézin-Gross-Witten tau-functions
- A Riemann Hilbert approach to the study of the generating function associated to the Pearcey process
- Universal Cusp Scaling in Random Partitions