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

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 (Gelfand–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.

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Soumis le : lundi 30 novembre 2020 - 15:21:05
Dernière modification le : jeudi 8 septembre 2022 - 15:08:23

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Mark Adler, Mattia Cafasso, Pierre van Moerbeke. Non-linear PDEs for gap probabilities in random matrices and KP theory. Physica D: Nonlinear Phenomena, Elsevier, 2012, 241 (23–24), pp.2265 - 2284. ⟨10.1016/j.physd.2012.08.016⟩. ⟨hal-03031613⟩



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