6,909 research outputs found
From Random Matrices to Stochastic Operators
We propose that classical random matrix models are properly viewed as finite
difference schemes for stochastic differential operators. Three particular
stochastic operators commonly arise, each associated with a familiar class of
local eigenvalue behavior. The stochastic Airy operator displays soft edge
behavior, associated with the Airy kernel. The stochastic Bessel operator
displays hard edge behavior, associated with the Bessel kernel. The article
concludes with suggestions for a stochastic sine operator, which would display
bulk behavior, associated with the sine kernel.Comment: 41 pages, 5 figures. Submitted to Journal of Statistical Physics.
Changes in this revision: recomputed Monte Carlo simulations, added reference
[19], fit into margins, performed minor editin
Random walks and random fixed-point free involutions
A bijection is given between fixed point free involutions of
with maximum decreasing subsequence size and two classes of vicious
(non-intersecting) random walker configurations confined to the half line
lattice points . In one class of walker configurations the maximum
displacement of the right most walker is . Because the scaled distribution
of the maximum decreasing subsequence size is known to be in the soft edge GOE
(random real symmetric matrices) universality class, the same holds true for
the scaled distribution of the maximum displacement of the right most walker.Comment: 10 page
Gap Probabilities for Edge Intervals in Finite Gaussian and Jacobi Unitary Matrix Ensembles
The probabilities for gaps in the eigenvalue spectrum of the finite dimension
random matrix Hermite and Jacobi unitary ensembles on some
single and disconnected double intervals are found. These are cases where a
reflection symmetry exists and the probability factors into two other related
probabilities, defined on single intervals. Our investigation uses the system
of partial differential equations arising from the Fredholm determinant
expression for the gap probability and the differential-recurrence equations
satisfied by Hermite and Jacobi orthogonal polynomials. In our study we find
second and third order nonlinear ordinary differential equations defining the
probabilities in the general case. For N=1 and N=2 the probabilities and
thus the solution of the equations are given explicitly. An asymptotic
expansion for large gap size is obtained from the equation in the Hermite case,
and also studied is the scaling at the edge of the Hermite spectrum as , and the Jacobi to Hermite limit; these last two studies make
correspondence to other cases reported here or known previously. Moreover, the
differential equation arising in the Hermite ensemble is solved in terms of an
explicit rational function of a {Painlev\'e-V} transcendent and its derivative,
and an analogous solution is provided in the two Jacobi cases but this time
involving a {Painlev\'e-VI} transcendent.Comment: 32 pages, Latex2
{\bf -Function Evaluation of Gap Probabilities in Orthogonal and Symplectic Matrix Ensembles}
It has recently been emphasized that all known exact evaluations of gap
probabilities for classical unitary matrix ensembles are in fact
-functions for certain Painlev\'e systems. We show that all exact
evaluations of gap probabilities for classical orthogonal matrix ensembles,
either known or derivable from the existing literature, are likewise
-functions for certain Painlev\'e systems. In the case of symplectic
matrix ensembles all exact evaluations, either known or derivable from the
existing literature, are identified as the mean of two -functions, both
of which correspond to Hamiltonians satisfying the same differential equation,
differing only in the boundary condition. Furthermore the product of these two
-functions gives the gap probability in the corresponding unitary
symmetry case, while one of those -functions is the gap probability in
the corresponding orthogonal symmetry case.Comment: AMS-Late
Effect of spin-orbit coupling on the excitation spectrum of Andreev billiards
We consider the effect of spin-orbit coupling on the low energy excitation
spectrum of an Andreev billiard (a quantum dot weakly coupled to a
superconductor), using a dynamical numerical model (the spin Andreev map).
Three effects of spin-orbit coupling are obtained in our simulations: In zero
magnetic field: (1) the narrowing of the distribution of the excitation gap;
(2) the appearance of oscillations in the average density of states. In strong
magnetic field: (3) the appearance of a peak in the average density of states
at zero energy. All three effects have been predicted by random-matrix theory.Comment: 5 pages, 4 figure
Two ways to solve ASEP
The purpose of this article is to describe the two approaches to compute
exact formulas (which are amenable to asymptotic analysis) for the probability
distribution of the current of particles past a given site in the asymmetric
simple exclusion process (ASEP) with step initial data. The first approach is
via a variant of the coordinate Bethe ansatz and was developed in work of Tracy
and Widom in 2008-2009, while the second approach is via a rigorous version of
the replica trick and was developed in work of Borodin, Sasamoto and the author
in 2012.Comment: 10 pages, Chapter in "Topics in percolative and disordered systems
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
Asymptotics of a discrete-time particle system near a reflecting boundary
We examine a discrete-time Markovian particle system on the quarter-plane
introduced by M. Defosseux. The vertical boundary acts as a reflecting wall.
The particle system lies in the Anisotropic Kardar-Parisi-Zhang with a wall
universality class. After projecting to a single horizontal level, we take the
longtime asymptotics and obtain the discrete Jacobi and symmetric Pearcey
kernels. This is achieved by showing that the particle system is identical to a
Markov chain arising from representations of the infinite-dimensional
orthogonal group. The fixed-time marginals of this Markov chain are known to be
determinantal point processes, allowing us to take the limit of the correlation
kernel.
We also give a simple example which shows that in the multi-level case, the
particle system and the Markov chain evolve differently.Comment: 16 pages, Version 2 improves the expositio
Edgeworth Expansion of the Largest Eigenvalue Distribution Function of GUE Revisited
We derive expansions of the resolvent
Rn(x;y;t)=(Qn(x;t)Pn(y;t)-Qn(y;t)Pn(x;t))/(x-y) of the Hermite kernel Kn at the
edge of the spectrum of the finite n Gaussian Unitary Ensemble (GUEn) and the
finite n expansion of Qn(x;t) and Pn(x;t). Using these large n expansions, we
give another proof of the derivation of an Edgeworth type theorem for the
largest eigenvalue distribution function of GUEn. We conclude with a brief
discussion on the derivation of the probability distribution function of the
corresponding largest eigenvalue in the Gaussian Orthogonal Ensemble (GOEn) and
Gaussian Symplectic Ensembles (GSEn)
Universal singularity at the closure of a gap in a random matrix theory
We consider a Hamiltonian , in which is a given
non-random Hermitian matrix,and is an Hermitian random matrix
with a Gaussian probability distribution.We had shown before that Dyson's
universality of the short-range correlations between energy levels holds at
generic points of the spectrum independently of . We consider here the
case in which the spectrum of is such that there is a gap in the
average density of eigenvalues of which is thus split into two pieces. When
the spectrum of is tuned so that the gap closes, a new class of
universality appears for the energy correlations in the vicinity of this
singular point.Comment: 20pages, Revtex, to be published in Phys. Rev.
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