Nonlinear 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 (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.
Cite
@article{arxiv.1104.4268,
title = {Nonlinear PDEs for gap probabilities in random matrices and KP theory},
author = {M. Adler and M. Cafasso and P. van Moerbeke},
journal= {arXiv preprint arXiv:1104.4268},
year = {2013}
}
Comments
Minor revision: accepted for publication on Physica D