Local universality of repulsive particle systems and random matrices

Abstract

We study local correlations of certain interacting particle systems on the real line which show repulsion similar to eigenvalues of random Hermitian matrices. Although the new particle system does not seem to have a natural spectral or determinantal representation, the local correlations in the bulk coincide in the limit of infinitely many particles with those known from random Hermitian matrices; in particular they can be expressed as determinants of the so-called sine kernel. These results may provide an explanation for the appearance of sine kernel correlation statistics in a number of situations which do not have an obvious interpretation in terms of random matrices.

Published in at http://dx.doi.org/10.1214/13-AOP844 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Fluctuations of eigenvalues of matrix models and their applications
• Bulk Universality of General $\beta$-Ensembles with Non-convex Potential
• Local universality of repulsive particle systems and random matrices
• Random matrices: The Four Moment Theorem for Wigner ensembles
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• Empirical Spacings of Unfolded Eigenvalues
• Particle Systems with Repulsion Exponent $\beta$ and Random Matrices
• Precise Deviations Results for the Maxima of Some Determinantal Point Processes: the Upper Tail