Spectral Universality of Real Chiral Random Matrix Ensembles

Abstract

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.

27 pages, 4 figures, Latex, corrected typos

References in corpus (5)

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• Character Expansions for the Orthogonal and Symplectic Groups
• Correlation Kernels for Discrete Symplectic and Orthogonal Ensembles
• Completing the picture for the smallest eigenvalue of real Wishart matrices
• Microscopic universality of complex matrix model correlation functions at weak non-Hermiticity
• New Critical Matrix Models and Generalized Universality
• Microscopic spectral density in random matrix models for chiral and diquark condensation
• QCD Anderson transition with overlap valence quarks on a twisted-mass sea
• Universal Distribution of Would-be Topological Zero Modes in Coupled Chiral Systems
• On the Microscopic Spectra of the Massive Dirac Operator for Chiral Orthogonal and Chiral Symplectic Ensembles