Macroscopic and Microscopic (Non-)Universality of Compact Support Random Matrix Theory
نویسنده
چکیده
A random matrix model with a σ-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 nonuniversal, extending previous results from monomial to arbitrary polynomial potentials. Using loop equation techniques we give a closed though nonuniversal expression for G(z,w), which extends recursively to all higher kpoint 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) = M2p, 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.
منابع مشابه
Determinantal point processes and random matrix theory in a nutshell
3 Universality 5 3.1 Macroscopic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.1.1 Wigner’s semicircle law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Microscopic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2.1 Bulk universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...
متن کاملNotes on Random Matrix Theory
3 Universality 4 3.1 Macroscopic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.1.1 Wigner’s semicircle law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Microscopic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2.1 Bulk universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...
متن کاملUniversality Limits at the Hard Edge of the Spectrum for Measures with Compact Support
We use the theory of entire functions and reproducing kernels to establish universality at the (hard) edge of the spectrum for a measure with compact support. This involves the Bessel kernel. In particular, we show that universality at the hard edge is equivalent to universality along the diagonal at the hard edge. 1. Results Let be a nite positive Borel measure with compact support supp[ ]. T...
متن کاملMicroscopic Spectra of Dirac Operators and Finite-Volume Partition Functions
Exact results from random matrix theory are used to systematically analyse the relationship between microscopic Dirac spectra and finite-volume partition functions. Results are presented for the unitary ensemble, and the chiral analogs of the three classical matrix ensembles: unitary, orthogonal and symplectic, all of which describe universality classes of SU(Nc) gauge theories with Nf fermions...
متن کاملO ct 1 99 8 Logarithmic Universality in Random Matrix The - ory
Universality in unitary invariant random matrix ensembles with complex matrix elements is considered. We treat two general ensembles which have a determinant factor in the weight. These ensembles are relevant, e.g., for spectra of the Dirac operator in QCD. In addition to the well established universality with respect to the choice of potential, we prove that microscopic spectral correlators ar...
متن کامل