Universal random matrix correlations of ratios of characteristic polynomials at the spectral edges

Abstract

It has been shown recently [10] that Cauchy transforms of orthogonal polynomials appear naturally in general correlation functions containing ratios of characteristic polynomials of random NxN Hermitian matrices. Our main goal is to investigate the issue of universality of large N asymptotics for those Cauchy transforms for a wide class of weight functions. Our analysis covers three different scaling regimes: the "hard edge", the "bulk" and the "soft edge" of the spectrum, thus extending the earlier results known for the bulk. The principal tool is to show that for finite matrix size N the auxiliary "wave functions" associated with the Cauchy transforms obey the same second order differential equation as those associated with the orthogonal polynomials themselves.

18 pages

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• Replica field theories, Painleve transcendents, and exact correlation functions
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• On matrix model partition functions for QCD with chemical potential
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• Comparison of the superbosonization formula and the generalized Hubbard-Stratonovich transformation
• On the asymptotics of some large Hankel determinants generated by Fisher-Hartwig symbols defined on the real line
• Asymptotics for products of characteristic polynomials in classical $\beta$-Ensembles
• Universality of the matrix Airy and Bessel functions at spectral edges of unitary ensembles
• Surprising Pfaffian factorizations in Random Matrix Theory with Dyson index $\beta =2$
• Bosonic Partition Functions at Nonzero (Imaginary) Chemical Potential
• Phase transitions for products of characteristic polynomials under Dyson Brownian motion