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Total integrals of global solutions to Painlevé II. (English) Zbl 1179.33036

The problem of finding asymptotic expansions of the distributions of the largest eigenvalue of the Gaussian orthogonal or Gaussian symplectic ensembles of random matrices up to constant terms employs the total integral (i.e., from \(x=-\infty\) to \(x=+\infty\)) of the special Hastings-McLeod solution of the second Painlevé equation with the parameter \(\alpha=0\) (increasing terms can be easily deduced from the asymptotics of the Painlevé function). The authors give a relatively short evaluation of the total integrals (regularized, if necessary) of all non-singular solutions on the real line to PII including real and imaginary Ablowitz-Segur solutions, Hastings-McLeod solution and increasing as \(x\to+\infty\) imaginary solutions (the latter integrals are computed modulo \(2\pi i\)). These and similar computation of the total integral of a function related to a special solution of the Painlevé V equation allow the authors to re-evaluate the constant terms in the asymptotic expansions of the limiting gap probabilities at the edge and the bulk for a GOE and GSE. Additionally, the authors compute the total integrals of special polynomials of the non-singular solutions to PII and their derivatives which can be thought of as analogs of the so-called trace formulae known in the soliton theory.
A conventional isomonodromy deformation approach to the asymptotics of the second Painlevé transcendent \(u(x)\) is based on the fact that this function appears in a leading non-trivial coefficient of the asymptotic expansion at \(\lambda=\infty\) for the solution \(\Psi(\lambda;x)\) of the associated Riemann-Hilbert problem. The authors observe that \(\Psi(\lambda;x)\) contains more interesting information. E.g., \(\Psi(\lambda;x)\) computed at \(\lambda=0\) involves an anti-derivative of \(u(x)\). Therefore comparing the values \(\Psi(0;-\infty)\) and \(\Psi(0;+\infty)\), it is possible to extract the value of the total integral of \(u(x)\). Similarly, the study of the higher-order terms of the expansion of \(\Psi(\lambda;x)\) at \(\lambda=\infty\) yields anti-derivatives of certain polynomials of \(u(x)\) and \(u'(x)\). This yields analogs of the above mentioned trace formulae.

MSC:

33E17 Painlevé-type functions
35Q15 Riemann-Hilbert problems in context of PDEs
15B52 Random matrices (algebraic aspects)
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