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Macroscopic and microscopic (non-)universality of compact support random matrix theory. (English) Zbl 0984.82028

Summary: 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.

MSC:

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics

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