Fredholm Determinant Solutions Of The Painleve Ii Hierarchy And Gap Probabilities Of Determinantal Point Processes

We study Fredholm determinants of a class of integral operators, whose kernels can be expressed as double contour integrals of a special type. Such Fredholm determinants appear in various random matrix and statistical physics models. We show that the logarithmic derivatives of the Fredholm determinants are directly related to solutions of the Painleve II hierarchy. This confirms and generalizes a recent conjecture by Le Doussal, Majumdar, and Schehr [20]. In addition, we obtain asymptotics at +/-infinity for the Painleve transcendents and large gap asymptotics for the corresponding point processes.