37 research outputs found
Characteristic polynomials of random matrices
Number theorists have studied extensively the connections between the
distribution of zeros of the Riemann -function, and of some
generalizations, with the statistics of the eigenvalues of large random
matrices. It is interesting to compare the average moments of these functions
in an interval to their counterpart in random matrices, which are the
expectation values of the characteristic polynomials of the matrix. It turns
out that these expectation values are quite interesting. For instance, the
moments of order 2K scale, for unitary invariant ensembles, as the density of
eigenvalues raised to the power ; the prefactor turns out to be a
universal number, i.e. it is independent of the specific probability
distribution. An equivalent behaviour and prefactor had been found, as a
conjecture, within number theory. The moments of the characteristic
determinants of random matrices are computed here as limits, at coinciding
points, of multi-point correlators of determinants. These correlators are in
fact universal in Dyson's scaling limit in which the difference between the
points goes to zero, the size of the matrix goes to infinity, and their product
remains finite.Comment: 30 pages,late
An extension of the HarishChandra-Itzykson-Zuber integral
The HarishChandra-Itzykson-Zuber integral over the unitary group U(k)
(beta=2) is present in numerous problems involving Hermitian random matrices.
It is well known that the result is semi-classically exact. This simple result
does not extend to other symmetry groups, such as the symplectic or orthogonal
groups. In this article the analysis of this integral is extended first to the
symplectic group Sp(k) (beta=4). There the semi-classical approximation has to
be corrected by a WKB expansion. It turns out that this expansion stops after a
finite number of terms ; in other words the WKB approximation is corrected by a
polynomial in the appropriate variables. The analysis is based upon new
solutions to the heat kernel differential equation. We have also investigated
arbitrary values of the parameter beta, which characterizes the symmetry group.
Closed formulae are derived for arbitrary beta and k=3, and also for large beta
and arbitrary k.Comment: 18 page
Characteristic polynomials of real symmetric random matrices
It is shown that the correlation functions of the random variables
, in which is a real symmetric random
matrix, exhibit universal local statistics in the large limit. The
derivation relies on an exact dual representation of the problem: the -point
functions are expressed in terms of finite integrals over (quaternionic)
matrices. However the control of the Dyson limit, in which the
distance of the various parameters \la's is of the order of the mean spacing,
requires an integration over the symplectic group. It is shown that a
generalization of the Itzykson-Zuber method holds for this problem, but
contrary to the unitary case, the semi-classical result requires a {\it finite}
number of corrections to be exact.
We have also considered the problem of an external matrix source coupled to
the random matrix, and obtain explicit integral formulae, which are useful for
the analysis of the large limit.Comment: 24 pages, late
Random super matrices with an external source
In the past we have considered Gaussian random matrix ensembles in the
presence of an external matrix source. The reason was that it allowed, through
an appropriate tuning of the eigenvalues of the source, to obtain results on
non-trivial dual models, such as Kontsevich's Airy matrix models and
generalizations. The techniques relied on explicit computations of the k-point
functions for arbitrary N (the size of the matrices) and on an N-k duality.
Numerous results on the intersection numbers of the moduli space of curves were
obtained by this technique. In order to generalize these results to include
surfaces with boundaries, we have extended these techniques to supermatrices.
Again we have obtained quite remarkable explicit expressions for the k-point
functions, as well as a duality. Although supermatrix models a priori lead to
the same matrix models of 2d-gravity, the external source extensions considered
in this article lead to new geometric results.Comment: 12 page
The intersection numbers of the p-spin curves from random matrix theory
The intersection numbers of p-spin curves are computed through correlation
functions of Gaussian ensembles of random matrices in an external matrix
source. The p-dependence of intersection numbers is determined as polynomial in
p; the large p behavior is also considered. The analytic continuation of
intersection numbers to negative values of p is discussed in relation to
SL(2,R)/U(1) black hole sigma model.Comment: 19 page
Level statistics inside the vortex of a superconductor and symplectic random matrix theory in an external source
In the core of the vortex of a superconductor, energy levels appear inside
the gap. We discuss here through a random matrix approach how these levels are
broadened by impurities. It is first shown that the level statistics is
governed by an ensemble consisting of a symplectic random potential added to a
non-random matrix. A generalization of previous work on the unitary ensemble in
the presence of an external source (which relied on the Itzykson-Zuber
integral) is discussed for this symplectic case through the formalism
introduced by Harish-Chandra and Duistermaat-Heckman. This leads to explicit
formulae for the density of states and for the correlation functions, which
describe the cross-over from the clean to the dirty limits.Comment: 34 pages, Revte
The Anderson Transition in Two-Dimensional Systems with Spin-Orbit Coupling
We report a numerical investigation of the Anderson transition in
two-dimensional systems with spin-orbit coupling. An accurate estimate of the
critical exponent for the divergence of the localization length in this
universality class has to our knowledge not been reported in the literature.
Here we analyse the SU(2) model. We find that for this model corrections to
scaling due to irrelevant scaling variables may be neglected permitting an
accurate estimate of the exponent