Constructive Approximation

, Volume 30, Issue 2, pp 225–263 | Cite as

First Colonization of a Spectral Outpost in Random Matrix Theory

Article

Abstract

We describe the distribution of the first finite number of eigenvalues in a newly-forming band of the spectrum of the random Hermitian matrix model. The method is rigorously based on the Riemann–Hilbert analysis of the corresponding orthogonal polynomials. We provide an analysis with an error term of order N−2γ where 1/γ=2ν+2 is the exponent of non-regularity of the effective potential, thus improving even in the usual case the analysis of the pertinent literature.

The behavior of the first finite number of zeroes (eigenvalues) appearing in the new band is analyzed and connected with the location of the zeroes of certain Freud polynomials. In general, all these newborn zeroes approach the point of nonregularity at the rate Nγ, whereas one (a stray zero) lags behind at a slower rate of approach. The kernels for the correlator functions in the scaling coordinate near the emerging band are provided together with the subleading term. In particular, the transition between K and K+1 eigenvalues is analyzed in detail.

Keywords

Orthogonal polynomials Random matrix theory Schlesinger transformations Riemann–Hilbert problems 

Mathematics Subject Classification (2000)

05E35 15A52 

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© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsConcordia UniversityMontréalCanada
  2. 2.Centre de recherches mathématiquesUniversité de MontréalMontréalCanada

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