17,263 research outputs found
Limiting Laws of Linear Eigenvalue Statistics for Unitary Invariant Matrix Models
We study the variance and the Laplace transform of the probability law of
linear eigenvalue statistics of unitary invariant Matrix Models of
n-dimentional Hermitian matrices as n tends to infinity. Assuming that the test
function of statistics is smooth enough and using the asymptotic formulas by
Deift et al for orthogonal polynomials with varying weights, we show first that
if the support of the Density of States of the model consists of two or more
intervals, then in the global regime the variance of statistics is a
quasiperiodic function of n generically in the potential, determining the
model. We show next that the exponent of the Laplace transform of the
probability law is not in general 1/2variance, as it should be if the Central
Limit Theorem would be valid, and we find the asymptotic form of the Laplace
transform of the probability law in certain cases
Random matrices with external source and KP functions
In this paper we prove that the partition function in the random matrix model
with external source is a KP function.Comment: 12 pages, title change
Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
We compute exact asymptotic results for the probability of the occurrence of
large deviations of the largest (smallest) eigenvalue of random matrices
belonging to the Gaussian orthogonal, unitary and symplectic ensembles. In
particular, we show that the probability that all the eigenvalues of an (NxN)
random matrix are positive (negative) decreases for large N as ~\exp[-\beta
\theta(0) N^2] where the Dyson index \beta characterizes the ensemble and the
exponent \theta(0)=(\ln 3)/4=0.274653... is universal. We compute the
probability that the eigenvalues lie in the interval [\zeta_1,\zeta_2] which
allows us to calculate the joint probability distribution of the minimum and
the maximum eigenvalue. As a byproduct, we also obtain exactly the average
density of states in Gaussian ensembles whose eigenvalues are restricted to lie
in the interval [\zeta_1,\zeta_2], thus generalizing the celebrated Wigner
semi-circle law to these restricted ensembles. It is found that the density of
states generically exhibits an inverse square-root singularity at the location
of the barriers. These results are confirmed by numerical simulations.Comment: 17 pages Revtex, 5 .eps figures include
The Index Distribution of Gaussian Random Matrices
We compute analytically, for large N, the probability distribution of the
number of positive eigenvalues (the index N_{+}) of a random NxN matrix
belonging to Gaussian orthogonal (\beta=1), unitary (\beta=2) or symplectic
(\beta=4) ensembles. The distribution of the fraction of positive eigenvalues
c=N_{+}/N scales, for large N, as Prob(c,N)\simeq\exp[-\beta N^2 \Phi(c)] where
the rate function \Phi(c), symmetric around c=1/2 and universal (independent of
), is calculated exactly. The distribution has non-Gaussian tails, but
even near its peak at c=1/2 it is not strictly Gaussian due to an unusual
logarithmic singularity in the rate function.Comment: 4 pages Revtex, 4 .eps figures include
Condensation of `composite bosons' in a rotating BEC
We provide evidence for several novel phases in the dilute limit of rotating
BECs. By exact calculation of wavefunctions and energies for small numbers of
particles, we show that the states near integer angular momentum per particle
are best considered condensates of composite entities, involving vortices and
atoms. We are led to this result by explicit comparison with a description
purely in terms of vortices. Several parallels with the fractional quantum Hall
effect emerge, including the presence of the Pfaffian state.Comment: 4 pages, Latex, 3 figure
Multi-Channel Transport in Disordered Medium under Generic Scattering Conditions
Our study of the evolution of transmission eigenvalues, due to changes in
various physical parameters in a disordered region of arbitrary dimensions,
results in a generalization of the celebrated DMPK equation. The evolution is
shown to be governed by a single complexity parameter which implies a deep
level of universality of transport phenomena through a wide range of disordered
regions. We also find that the interaction among eigenvalues is of many body
type that has important consequences for the statistical behavior of transport
properties.Comment: 19 Pages, No Figure
Distributions of Conductance and Shot Noise and Associated Phase Transitions
For a chaotic cavity with two indentical leads each supporting N channels, we
compute analytically, for large N, the full distribution of the conductance and
the shot noise power and show that in both cases there is a central Gaussian
region flanked on both sides by non-Gaussian tails. The distribution is weakly
singular at the junction of Gaussian and non-Gaussian regimes, a direct
consequence of two phase transitions in an associated Coulomb gas problem.Comment: 5 pages, 3 figures include
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