Fredholm determinants of general (1,p)-kernels and reductions of non-linear integrable PDE's

Details Fredholm determinants of general (1,p)-kernels and reductions of non-linear integrable PDE's Fredholm determinants of general (1,p)-kernels and reductions of non-linear integrable PDE's

2011-04-21

arxiv.org/abs/1104.4268v1

Physics

Mathematical Physics

48 pages, 1 figure

Scientific paper

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 PDE's (Virasoro constraints), yielding, in a systematic way, non-linear PDE's 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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