9 research outputs found
Higher Order Analogues of Tracy-Widom Distributions via the Lax Method
We study the distribution of the largest eigenvalue in formal Hermitian
one-matrix models at multicriticality, where the spectral density acquires an
extra number of k-1 zeros at the edge. The distributions are directly expressed
through the norms of orthogonal polynomials on a semi-infinite interval, as an
alternative to using Fredholm determinants. They satisfy non-linear recurrence
relations which we show form a Lax pair, making contact to the string
literature in the early 1990's. The technique of pseudo-differential operators
allows us to give compact expressions for the logarithm of the gap probability
in terms of the Painleve XXXIV hierarchy. These are the higher order analogues
of the Tracy-Widom distribution which has k=1. Using known Backlund
transformations we show how to simplify earlier equivalent results that are
derived from Fredholm determinant theory, valid for even k in terms of the
Painleve II hierarchy.Comment: 24 pages. Improved discussion of Backlund transformations, in
addition to other minor improvements in text. Typos corrected. Matches
published versio
Purity distribution for generalized random Bures mixed states
We compute the distribution of the purity for random density matrices
(i.e.random mixed states) in a large quantum system, distributed according to
the Bures measure. The full distribution of the purity is computed using a
mapping to random matrix theory and then a Coulomb gas method. We find three
regimes that correspond to two phase transitions in the associated Coulomb gas.
The first transition is characterized by an explosion of the third derivative
on the left of the transition point. The second transition is of first order,
it is characterized by the detachement of a single charge of the Coulomb gas. A
key remark in this paper is that the random Bures states are closely related to
the O(n) model for n=1. This actually led us to study "generalized Bures
states" by keeping general instead of specializing to n=1
Large deviations of the maximal eigenvalue of random matrices
We present detailed computations of the 'at least finite' terms (three
dominant orders) of the free energy in a one-cut matrix model with a hard edge
a, in beta-ensembles, with any polynomial potential. beta is a positive number,
so not restricted to the standard values beta = 1 (hermitian matrices), beta =
1/2 (symmetric matrices), beta = 2 (quaternionic self-dual matrices). This
model allows to study the statistic of the maximum eigenvalue of random
matrices. We compute the large deviation function to the left of the expected
maximum. We specialize our results to the gaussian beta-ensembles and check
them numerically. Our method is based on general results and procedures already
developed in the literature to solve the Pastur equations (also called "loop
equations"). It allows to compute the left tail of the analog of Tracy-Widom
laws for any beta, including the constant term.Comment: 62 pages, 4 figures, pdflatex ; v2 bibliography corrected ; v3 typos
corrected and preprint added ; v4 few more numbers adde
Spectral density asymptotics for Gaussian and Laguerre -ensembles in the exponentially small region
The first two terms in the large asymptotic expansion of the
moment of the characteristic polynomial for the Gaussian and Laguerre
-ensembles are calculated. This is used to compute the asymptotic
expansion of the spectral density in these ensembles, in the exponentially
small region outside the leading support, up to terms . The leading form
of the right tail of the distribution of the largest eigenvalue is given by the
density in this regime. It is demonstrated that there is a scaling from this,
to the right tail asymptotics for the distribution of the largest eigenvalue at
the soft edge.Comment: 19 page
A simple derivation of the Tracy-Widom distribution of the maximal eigenvalue of a Gaussian unitary random matrix
In this paper, we first briefly review some recent results on the
distribution of the maximal eigenvalue of a random matrix drawn
from Gaussian ensembles. Next we focus on the Gaussian Unitary Ensemble (GUE)
and by suitably adapting a method of orthogonal polynomials developed by Gross
and Matytsin in the context of Yang-Mills theory in two dimensions, we provide
a rather simple derivation of the Tracy-Widom law for GUE. Our derivation is
based on the elementary asymptotic scaling analysis of a pair of coupled
nonlinear recursion relations. As an added bonus, this method also allows us to
compute the precise subleading terms describing the right large deviation tail
of the maximal eigenvalue distribution. In the Yang-Mills language, these
subleading terms correspond to non-perturbative (in expansion)
corrections to the two-dimensional partition function in the so called `weak'
coupling regime.Comment: 2 figure