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

Akemann G, Vernizzi G (2000)

Nucl.Phys. B 583(3): 739-757.

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Akemann, Gernot

^{UniBi}; Vernizzi, G.Abstract

A random matrix model with a sigma-model like constraint, the restrictedtrace ensemble (RTE), is solved in the large-n limit. In the macroscopic limitthe 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 expressionfor G(z,w), which extends recursively to all higher k-point resolvents. Thesefindings are in contrast to the usual unconstrained one-matrix model. However,in the microscopic large-n limit, which probes only correlations at distance ofthe mean level spacing, we are able to show that the constraint does not modifythe universal sine-law. In the case of monomial potentials $V(M)=M^{2p}$, weprovide a relation valid for finite-n between the k-point correlation functionof the RTE and the unconstrained model. In the microscopic large-n limit theycoincide which proves the microscopic universality of RTEs.

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Akemann G, Vernizzi G. Macroscopic and Microscopic (Non-)Universality of Compact Support Random Matrix Theory.

*Nucl.Phys. B*. 2000;583(3):739-757.Akemann, G., & Vernizzi, G. (2000). Macroscopic and Microscopic (Non-)Universality of Compact Support Random Matrix Theory.

*Nucl.Phys. B*,*583*(3), 739-757.Akemann, G., and Vernizzi, G. (2000). Macroscopic and Microscopic (Non-)Universality of Compact Support Random Matrix Theory.

*Nucl.Phys. B*583, 739-757.Akemann, G., & Vernizzi, G., 2000. Macroscopic and Microscopic (Non-)Universality of Compact Support Random Matrix Theory.

*Nucl.Phys. B*, 583(3), p 739-757. G. Akemann and G. Vernizzi, “Macroscopic and Microscopic (Non-)Universality of Compact Support Random Matrix Theory”,

*Nucl.Phys. B*, vol. 583, 2000, pp. 739-757. Akemann, G., Vernizzi, G.: Macroscopic and Microscopic (Non-)Universality of Compact Support Random Matrix Theory. Nucl.Phys. B. 583, 739-757 (2000).

Akemann, Gernot, and Vernizzi, G. “Macroscopic and Microscopic (Non-)Universality of Compact Support Random Matrix Theory”.

*Nucl.Phys. B*583.3 (2000): 739-757.
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arXiv hep-th/0002148