1,948 research outputs found
Chiral Random Matrix Model for Critical Statistics
We propose a random matrix model that interpolates between the chiral random
matrix ensembles and the chiral Poisson ensemble. By mapping this model on a
non-interacting Fermi-gas we show that for energy differences less than a
critical energy the spectral correlations are given by chiral Random
Matrix Theory whereas for energy differences larger than the number
variance shows a linear dependence on the energy difference with a slope that
depends on the parameters of the model. If the parameters are scaled such that
the slope remains fixed in the thermodynamic limit, this model provides a
description of QCD Dirac spectra in the universality class of critical
statistics. In this way a good description of QCD Dirac spectra for gauge field
configurations given by a liquid of instantons is obtained.Comment: 21 pages, 3 figures, Latex; added two references and minor
correction
Universal correlations in random matrices: quantum chaos, the integrable model, and quantum gravity
Random matrix theory (RMT) provides a common mathematical formulation of
distinct physical questions in three different areas: quantum chaos, the 1-d
integrable model with the interaction (the Calogero-Sutherland-Moser
system), and 2-d quantum gravity. We review the connection of RMT with these
areas. We also discuss the method of loop equations for determining correlation
functions in RMT, and smoothed global eigenvalue correlators in the 2-matrix
model for gaussian orthogonal, unitary and symplectic ensembles.Comment: 26 pages, LaTe
Random Matrix Theories in Quantum Physics: Common Concepts
We review the development of random-matrix theory (RMT) during the last
decade. We emphasize both the theoretical aspects, and the application of the
theory to a number of fields. These comprise chaotic and disordered systems,
the localization problem, many-body quantum systems, the Calogero-Sutherland
model, chiral symmetry breaking in QCD, and quantum gravity in two dimensions.
The review is preceded by a brief historical survey of the developments of RMT
and of localization theory since their inception. We emphasize the concepts
common to the above-mentioned fields as well as the great diversity of RMT. In
view of the universality of RMT, we suggest that the current development
signals the emergence of a new "statistical mechanics": Stochasticity and
general symmetry requirements lead to universal laws not based on dynamical
principles.Comment: 178 pages, Revtex, 45 figures, submitted to Physics Report
Spectral Properties of Wigner Matrices
In these notes we review recent progress (and, in Section \ref{sec:ados}, we
announce a new result) concerning the statistical properties of the spectrum of
Wigner random matrices.Comment: 14 pages, contribution to the Proceedings of the Conference QMath 11
held in Hradec Kralove (Czechia) in September 201
A note on biorthogonal ensembles
We consider ensembles of random matrices, known as biorthogonal ensembles,
whose eigenvalue probability density function can be written as a product of
two determinants. These systems are closely related to multiple orthogonal
functions. It is known that the eigenvalue correlation functions of such
ensembles can be written as a determinant of a kernel function. We show that
the kernel is itself an average of a single ratio of characteristic
polynomials. In the same vein, we prove that the type I multiple polynomials
can be expressed as an average of the inverse of a characteristic polynomial.
We finally introduce a new biorthogonal matrix ensemble, namely the chiral
unitary perturbed by a source term.Comment: 20 page
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