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

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.

LaTeX, 24 pages; Intro extended, 3 Refs added and typos corrected, to appear in NPB

References in corpus (7)

• Random-Matrix Theory of Quantum Transport
• Random Matrix Theories in Quantum Physics: Common Concepts
• Compact support probability distributions in random matrix theory
• Universal correlators for multi-arc complex matrix models
• Universality in invariant random-matrix models: Existence near the soft edge
• Non-universality of compact support probability distributions in random matrix theory
• Breakdown of Universality in Random Matrix Models

Cited by in corpus (12)

• Factorization of Correlation Functions and the Replica Limit of the Toda Lattice Equation
• Universality at weak and strong non-Hermiticity beyond the elliptic Ginibre ensemble
• Microscopic correlations of non-Hermitian Dirac operators in three-dimensional QCD
• Power-law deformation of Wishart-Laguerre ensembles of random matrices
• Compact smallest eigenvalue expressions in Wishart-Laguerre ensembles with or without fixed-trace
• Norm-dependent Random Matrix Ensembles in External Field and Supersymmetry
• New Critical Matrix Models and Generalized Universality
• Some Universal Properties for Restricted Trace Gaussian Orthogonal, Unitary and Symplectic Ensembles
• Products of random matrices from fixed trace and induced Ginibre ensembles
• Fixed trace $\beta$-Hermite ensembles: Asymptotic eigenvalue density and the edge of the density
• Invariant sums of random matrices and the onset of level repulsion
• Limit Correlation Functions for Fixed Trace Random Matrix Ensembles