Total integrals of global solutions to Painleve II

TL;DR

This paper computes total integrals of global solutions to Painleve II and related functions, providing new proofs and connections to random matrix theory, integrable systems, and classical trace formulae.

Contribution

It introduces novel methods to evaluate total integrals of Painleve II solutions and related functions, linking them to random matrix gap probabilities and classical spectral theory.

Findings

Total integrals of Painleve II solutions are explicitly evaluated.

New proofs for gap probability constants in Gaussian ensembles are provided.

Connections to trace formulae for the Dirac operator are established.

Abstract

We evaluate the total integral from negative infinity to positive infinity of all global solutions to the Painleve II equation on the real line. The method is based on the interplay between one of the equations of the associated Lax pair and the corresponding Riemann-Hilbert problem. In addition, we evaluate the total integral of a function related to a special solution to the Painleve V equation. As a corollary, we obtain short proofs of the computation of the constant terms of the limiting gap probabilities in the edge and the bulk of the Gaussian Orthogonal and Gaussian Symplectic Ensembles that were obtained recently in [4] and [18]. We also evaluate the total integrals of certain polynomials of the Painleve functions and their derivatives. These polynomials are the densities of the first integrals of the modified Korteweg-de Vries equation. We discuss the relations of the formulae we have obtained to the classical trace formulae for the Dirac operator on the line.