1,431 research outputs found
Logarithmic moments of characteristic polynomials of random matrices
In a recent article we have discussed the connections between averages of
powers of Riemann's -function on the critical line, and averages of
characteristic polynomials of random matrices. The result for random matrices
was shown to be universal, i.e. independent of the specific probability
distribution, and the results were derived for arbitrary moments. This allows
one to extend the previous results to logarithmic moments, for which we derive
the explicit universal expressions in random matrix theory. We then compare
these results to various results and conjectures for -functions, and the
correspondence is again striking.Comment: 10 pages, late
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]
Perturbative analysis of an n-Ising model on a random surface
Two dimensional quantum gravity coupled to a conformally invariant matter
field of central charge c=n/2, is represented, in a discretized version, by n
independent Ising spins per cell of the triangulations of a random surface. The
matrix integral representation of this model leads to a diagrammatic expansion
at large orders, when the Ising coupling constant is tuned to criticality, one
extracts the values of the string susceptibility exponent. We extend our
previous calculation to order eight for genus zero and investigate now also the
genus one case in order to check the possibility of having a well-defined
double scaling limit even c>1.Comment: 9p
Universal Wide Correlators in Non-Gaussian Orthogonal, Unitary and Symplectic Random Matrix Ensembles
We calculate wide distance connected correlators in non-gaussian orthogonal,
unitary and symplectic random matrix ensembles by solving the loop equation in
the 1/N-expansion. The multi-level correlator is shown to be universal in large
N limit. We show the algorithm to obtain the connected correlator to an
arbitrary order in the 1/N-expansion.Comment: 10 pages LaTex, Some minor changes have been mad
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]
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
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
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
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
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