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

Abstract

It has been shown recently by Fyodorov and Strahov [math-ph/0204051] that Cauchy transforms of orthogonal polynomials appear naturally in general correlation functions containing ratios of characteristic polynomials of random N×N 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

Bosonic partition functions at nonzero (imaginary) chemical potential

“…This is indeed the Cauchy transform of a Laguerre polynomial [44], which is the correct finite n result for the chiral random matrix partition function. For m → 0 we have that…”

Universal Behavior for Averages of Characteristic Polynomials at the Origin of the Spectrum

“…This scaling limit is called the origin of the spectrum by various authors, see for example [4,6,20,25]. It will turn out that this universal behavior is described in terms of the Bessel kernels given in Table 2.…”

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