78 research outputs found
Parametric level statistics in random matrix theory: Exact solution
An exact solution to the problem of parametric level statistics in
non-Gaussian ensembles of N by N Hermitian random matrices with either soft or
strong level confinement is formulated within the framework of the orthogonal
polynomial technique. Being applied to random matrices with strong level
confinement, the solution obtained leads to emergence of a new connection
relation that makes a link between the parametric level statistics and the
scalar two-point kernel in the thermodynamic limit.Comment: 4 pages (revtex
A Note on the Pfaffian Integration Theorem
Two alternative, fairly compact proofs are presented of the Pfaffian
integration theorem that is surfaced in the recent studies of spectral
properties of Ginibre's Orthogonal Ensemble. The first proof is based on a
concept of the Fredholm Pfaffian; the second proof is purely linear-algebraic.Comment: 8 pages; published versio
Random matrices and the replica method
Recent developments [Kamenev and Mezard, cond-mat/9901110, cond-mat/9903001;
Yurkevich and Lerner, cond-mat/9903025; Zirnbauer, cond-mat/9903338] have
revived a discussion about applicability of the replica approach to description
of spectral fluctuations in the context of random matrix theory and beyond. The
present paper, concentrating on invariant non-Gaussian random matrix ensembles
with orthogonal, unitary and symplectic symmetries, aims to demonstrate that
both the bosonic and the fermionic replicas are capable of reproducing
nonperturbative fluctuation formulas for spectral correlation functions in
entire energy scale, including the self-correlation of energy levels, provided
no sigma-model mapping is used.Comment: 12 pages (latex), presentation clarified, misprints fixe
Random matrix models with log-singular level confinement: method of fictitious fermions
Joint distribution function of N eigenvalues of U(N) invariant random-matrix
ensemble can be interpreted as a probability density to find N fictitious
non-interacting fermions to be confined in a one-dimensional space. Within this
picture a general formalism is developed to study the eigenvalue correlations
in non-Gaussian ensembles of large random matrices possessing non-monotonic,
log-singular level confinement. An effective one-particle Schroedinger equation
for wave-functions of fictitious fermions is derived. It is shown that
eigenvalue correlations are completely determined by the Dyson's density of
states and by the parameter of the logarithmic singularity. Closed analytical
expressions for the two-point kernel in the origin, bulk, and soft-edge scaling
limits are deduced in a unified way, and novel universal correlations are
predicted near the end point of the single spectrum support.Comment: 13 pages (latex), Presented at the MINERVA Workshop on Mesoscopics,
Fractals and Neural Networks, Eilat, Israel, March 199
Eigenvalue statistics of the real Ginibre ensemble
The real Ginibre ensemble consists of random matrices formed
from i.i.d. standard Gaussian entries. By using the method of skew orthogonal
polynomials, the general -point correlations for the real eigenvalues, and
for the complex eigenvalues, are given as Pfaffians with explicit
entries. A computationally tractable formula for the cumulative probability
density of the largest real eigenvalue is presented. This is relevant to May's
stability analysis of biological webs.Comment: 4 pages, to appear PR
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