790 research outputs found
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 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
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
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]
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
Computing topological invariants with one and two-matrix models
A generalization of the Kontsevich Airy-model allows one to compute the
intersection numbers of the moduli space of p-spin curves. These models are
deduced from averages of characteristic polynomials over Gaussian ensembles of
random matrices in an external matrix source. After use of a duality, and of an
appropriate tuning of the source, we obtain in a double scaling limit these
intersection numbers as polynomials in p. One can then take the limit p to -1
which yields a matrix model for orbifold Euler characteristics. The
generalization to a time-dependent matrix model, which is equivalent to a
two-matrix model, may be treated along the same lines ; it also yields a
logarithmic potential with additional vertices for general p.Comment: 30 pages, added references, changed conten
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
Universal correlations for deterministic plus random Hamiltonians
We consider the (smoothed) average correlation between the density of energy
levels of a disordered system, in which the Hamiltonian is equal to the sum of
a deterministic H0 and of a random potential . Remarkably, this
correlation function may be explicitly determined in the limit of large
matrices, for any unperturbed H0 and for a class of probability distribution
P of the random potential. We find a compact representation of the
correlation function. From this representation one obtains readily the short
distance behavior, which has been conjectured in various contexts to be
universal. Indeed we find that it is totally independent of both H0 and
P().Comment: 26P, (+5 figures not included
Correlations between eigenvalues of large random matrices with independent entries
We derive the connected correlation functions for eigenvalues of large
Hermitian random matrices with independently distributed elements using both a
diagrammatic and a renormalization group (RG) inspired approach. With the
diagrammatic method we obtain a general form for the one, two and three-point
connected Green function for this class of ensembles when matrix elements are
identically distributed, and then discuss the derivation of higher order
functions by the same approach. Using the RG approach we re-derive the one and
two-point Green functions and show they are unchanged by choosing certain
ensembles with non-identically distributed elements. Throughout, we compare the
Green functions we obtain to those from the class of ensembles with unitary
invariant distributions and discuss universality in both ensemble classes.Comment: 23 pages, RevTex, hard figures available from [email protected]
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
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