723 research outputs found
Universal relation between Green's functions in random matrix theory
We prove that in random matrix theory there exists a universal relation
between the one-point Green's function and the connected two- point Green's
function given by \vfill N^2 G_c(z,w) = {\part^2 \over \part z \part w}
\log (({G(z)- G(w) \over z -w}) + {\rm {irrelevant \ factorized \ terms.}}
This relation is universal in the sense that it does not depend on the
probability distribution of the random matrices for a broad class of
distributions, even though is known to depend on the probability
distribution in detail. The universality discussed here represents a different
statement than the universality we discovered a couple of years ago, which
states that is independent of the probability distribution,
where denotes the width of the spectrum and depends sensitively on the
probability distribution. It is shown that the universality proved here also
holds for the more general problem of a Hamiltonian consisting of the sum of a
deterministic term and a random term analyzed perturbatively by Br\'ezin,
Hikami, and Zee.Comment: 34 pages, macros appended (shorts, defs, boldchar), hard figures or
PICT figure files available from: [email protected]
Law of addition in random matrix theory
We discuss the problem of adding random matrices, which enable us to study
Hamiltonians consisting of a deterministic term plus a random term. Using a
diagrammatic approach and introducing the concept of ``gluon connectedness," we
calculate the density of energy levels for a wide class of probability
distributions governing the random term, thus generalizing a result obtained
recently by Br\'ezin, Hikami, and Zee. The method used here may be applied to a
broad class of problems involving random matrices.Comment: 17 pages, Latex with special macro appended, hard figs available
from: [email protected]
Universal Spectral Correlation between Hamiltonians with Disorder
We study the correlation between the energy spectra of two disordered
Hamiltonians of the form () with and
drawn from random distributions. We calculate this correlation
function explicitly and show that it has a simple universal form for a broad
class of random distributions.Comment: 9 pages, Jnl.tex Version 0.3 (version taken from the bulletin board),
NSF-ITP-93-13
On an Airy matrix model with a logarithmic potential
The Kontsevich-Penner model, an Airy matrix model with a logarithmic
potential, may be derived from a simple Gaussian two-matrix model through a
duality. In this dual version the Fourier transforms of the n-point correlation
functions can be computed in closed form. Using Virasoro constraints, we find
that in addition to the parameters , which appears in the KdV hierarchies,
one needs to introduce here half-integer indices .
The free energy as a function of those parameters may be obtained from these
Virasoro constraints. The large N limit follows from the solution to an
integral equation. This leads to explicit computations for a number of
topological invariants.Comment: 35 page
Intersection theory from duality and replica
Kontsevich's work on Airy matrix integrals has led to explicit results for
the intersection numbers of the moduli space of curves. In this article we show
that a duality between k-point functions on matrices and N-point
functions of matrices, plus the replica method, familiar in the
theory of disordered systems, allows one to recover Kontsevich's results on the
intersection numbers, and to generalize them to other models. This provides an
alternative and simple way to compute intersection numbers with one marked
point, and leads also to some new results
Intersection numbers of Riemann surfaces from Gaussian matrix models
We consider a Gaussian random matrix theory in the presence of an external
matrix source. This matrix model, after duality (a simple version of the
closed/open string duality), yields a generalized Kontsevich model through an
appropriate tuning of the external source. The n-point correlation functions of
this theory are shown to provide the intersection numbers of the moduli space
of curves with a p-spin structure, n marked points and top Chern class. This
sheds some light on Witten's conjecture on the relationship with the pth-KdV
equation
Oscillating density of states near zero energy for matrices made of blocks with possible application to the random flux problem
We consider random hermitian matrices made of complex blocks. The symmetries
of these matrices force them to have pairs of opposite real eigenvalues, so
that the average density of eigenvalues must vanish at the origin. These
densities are studied for finite matrices in the Gaussian ensemble.
In the large limit the density of eigenvalues is given by a semi-circle
law. However, near the origin there is a region of size in which
this density rises from zero to the semi-circle, going through an oscillatory
behavior. This cross-over is calculated explicitly by various techniques. We
then show to first order in the non-Gaussian character of the probability
distribution that this oscillatory behavior is universal, i.e. independent of
the probability distribution. We conjecture that this universality holds to all
orders. We then extend our consideration to the more complicated block matrices
which arise from lattices of matrices considered in our previous work. Finally,
we study the case of random real symmetric matrices made of blocks. By using a
remarkable identity we are able to determine the oscillatory behavior in this
case also. The universal oscillations studied here may be applicable to the
problem of a particle propagating on a lattice with random magnetic flux.Comment: 47 pages, regular LateX, no figure
An Extension of Level-spacing Universality
Dyson's short-distance universality of the correlation functions implies the
universality of P(s), the level-spacing distribution. We first briefly review
how this property is understood for unitary invariant ensembles and consider
next a Hamiltonian H = H_0+ V , in which H_0 is a given, non-random, N by N
matrix, and V is an Hermitian random matrix with a Gaussian probability
distribution. n-point correlation function may still be expressed as a
determinant of an n by n matrix, whose elements are given by a kernel
as in the H_0=0 case. From this representation we can show
that Dyson's short-distance universality still holds. We then conclude that
P(s) is independent of H_0.Comment: 12 pages, Revte
Characteristic polynomials of random matrices at edge singularities
We have discussed earlier the correlation functions of the random variables
\det(\la-X) in which is a random matrix. In particular the moments of the
distribution of these random variables are universal functions, when measured
in the appropriate units of the level spacing. When the \la's, instead of
belonging to the bulk of the spectrum, approach the edge, a cross-over takes
place to an Airy or to a Bessel problem, and we consider here these modified
classes of universality.
Furthermore, when an external matrix source is added to the probability
distribution of , various new phenomenons may occur and one can tune the
spectrum of this source matrix to new critical points. Again there are
remarkably simple formulae for arbitrary source matrices, which allow us to
compute the moments of the characteristic polynomials in these cases as well.Comment: 22 pages, late
Universal singularity at the closure of a gap in a random matrix theory
We consider a Hamiltonian , in which is a given
non-random Hermitian matrix,and is an Hermitian random matrix
with a Gaussian probability distribution.We had shown before that Dyson's
universality of the short-range correlations between energy levels holds at
generic points of the spectrum independently of . We consider here the
case in which the spectrum of is such that there is a gap in the
average density of eigenvalues of which is thus split into two pieces. When
the spectrum of is tuned so that the gap closes, a new class of
universality appears for the energy correlations in the vicinity of this
singular point.Comment: 20pages, Revtex, to be published in Phys. Rev.
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