Fredholm determinants and pole-free solutions to the noncommutative Painleve' II equation

Abstract

We extend the formalism of integrable operators a' la Its-Izergin-Korepin-Slavnov to matrix-valued convolution operators on a semi-infinite interval and to matrix integral operators with a kernel of the form E_1^T(x) E_2(y)/(x+y) thus proving that their resolvent operators can be expressed in terms of solutions of some specific Riemann-Hilbert problems. We also describe some applications, mainly to a noncommutative version of Painleve' II (recently introduced by Retakh and Rubtsov), a related noncommutative equation of Painleve' type. We construct a particular family of solutions of the noncommutative Painleve' II that are pole-free (for real values of the variables) and hence analogous to the Hastings-McLeod solution of (commutative) Painleve' II. Such a solution plays the same role as its commutative counterpart relative to the Tracy-Widom theorem, but for the computation of the Fredholm determinant of a matrix version of the Airy kernel.

46 pages, no figures (oddly)

References in corpus (2)

• A determinantal formula for the GOE Tracy-Widom distribution
• Noncommutative Toda Chains, Hankel Quasideterminants And Painlev'e II Equation

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• Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane
• Noncommutative Painlevé equations and systems of Calogero type
• Non-commutative q-Painleve VI equation
• The Toda and Painlevé Systems Associated with Semiclassical Matrix-Valued Orthogonal Polynomials of Laguerre Type
• A fully noncommutative Painlevé II hierarchy: Lax pair and solutions related to Fredholm determinants