Spectral Universality of Real Chiral Random Matrix Ensembles

Details Spectral Universality of Real Chiral Random Matrix Ensembles Spectral Universality of Real Chiral Random Matrix Ensembles

2000-04-17

arxiv.org/abs/hep-th/0004119v2

Nucl.Phys.B588:483-507,2000

Physics

High Energy Physics

High Energy Physics - Theory

27 pages, 4 figures, Latex, corrected typos

Scientific paper

10.1016/S0550-3213(00)00448-X

We investigate the universality of microscopic eigenvalue correlations for Random Matrix Theories with the global symmetries of the QCD partition function. In this article we analyze the case of real valued chiral Random Matrix Theories ($\beta =1$) by relating the kernel of the correlations functions for $\beta =1$ to the kernel of chiral Random Matrix Theories with complex matrix elements ($\beta = 2$), which is already known to be universal. Our proof is based on a novel asymptotic property of the skew-orthogonal polynomials: an integral over the corresponding wavefunctions oscillates about half its asymptotic value in the region of the bulk of the zeros. This result solves the puzzle that microscopic universality persists in spite of contributions to the microscopic correlators from the region near the largest zero of the skew-orthogonal polynomials. Our analytical results are illustrated by the numerical construction of the skew-orthogonal polynomials for an $x^4$ probability potential.

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