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Macroscopic and Microscopic (Non-)Universality of Compact Support Random Matrix Theory

G. Akemann·G. Vernizzi
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Abstract

A random matrix model with a sigma-model like constraint, the restricted trace ensemble (RTE), is solved in the large-n limit. In the macroscopic limit the smooth connected two-point resolvent G(z,w) is found to be non-universal, extending previous results from monomial to arbitrary polynomial potentials. Using loop equation techniques we give a closed though non-universal expression for G(z,w), which extends recursively to all higher k-point resolvents. These findings are in contrast to the usual unconstrained one-matrix model. However, in the microscopic large-n limit, which probes only correlations at distance of the mean level spacing, we are able to show that the constraint does not modify the universal sine-law. In the case of monomial potentials $V(M)=M^{2p}$, we provide a relation valid for finite-n between the k-point correlation function of the RTE and the unconstrained model. In the microscopic large-n limit they coincide which proves the microscopic universality of RTEs.

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G. Akemann, G. Vernizzi. 2000-06-07. Macroscopic and Microscopic (Non-)Universality of Compact Support Random Matrix Theory. https://doi.org/10.1016/s0550-3213(00)00325-4

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