6,159 research outputs found
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
{\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
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
Increasing subsequences and the hard-to-soft edge transition in matrix ensembles
Our interest is in the cumulative probabilities Pr(L(t) \le l) for the
maximum length of increasing subsequences in Poissonized ensembles of random
permutations, random fixed point free involutions and reversed random fixed
point free involutions. It is shown that these probabilities are equal to the
hard edge gap probability for matrix ensembles with unitary, orthogonal and
symplectic symmetry respectively. The gap probabilities can be written as a sum
over correlations for certain determinantal point processes. From these
expressions a proof can be given that the limiting form of Pr(L(t) \le l) in
the three cases is equal to the soft edge gap probability for matrix ensembles
with unitary, orthogonal and symplectic symmetry respectively, thereby
reclaiming theorems due to Baik-Deift-Johansson and Baik-Rains.Comment: LaTeX, 19 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
Nonintersecting Brownian motions on the half-line and discrete Gaussian orthogonal polynomials
We study the distribution of the maximal height of the outermost path in the
model of nonintersecting Brownian motions on the half-line as , showing that it converges in the proper scaling to the Tracy-Widom
distribution for the largest eigenvalue of the Gaussian orthogonal ensemble.
This is as expected from the viewpoint that the maximal height of the outermost
path converges to the maximum of the process minus a
parabola. Our proof is based on Riemann-Hilbert analysis of a system of
discrete orthogonal polynomials with a Gaussian weight in the double scaling
limit as this system approaches saturation. We consequently compute the
asymptotics of the free energy and the reproducing kernel of the corresponding
discrete orthogonal polynomial ensemble in the critical scaling in which the
density of particles approaches saturation. Both of these results can be viewed
as dual to the case in which the mean density of eigenvalues in a random matrix
model is vanishing at one point.Comment: 39 pages, 4 figures; The title has been changed from "The limiting
distribution of the maximal height of nonintersecting Brownian excursions and
discrete Gaussian orthogonal polynomials." This is a reflection of the fact
that the analysis has been adapted to include nonintersecting Brownian
motions with either reflecting of absorbing boundaries at zero. To appear in
J. Stat. Phy
Resistively-Detected NMR in a Two-Dimensional Electron System near : Clues to the Origin of the Dispersive Lineshape
Resistively-detected NMR measurements on 2D electron systems near the quantum Hall state are reported. In contrast to recent results of Gervais
\emph{et al.} [Phys. Rev. Lett. , 196803 (2005)], a dispersive
lineshape is found at all RF powers studied and Korringa-like nuclear
spin-lattice relaxation is observed. The shape of the unexplained dispersive
lineshape is found to invert when the temperature derivative of the
longitudinal resistance changes sign. This suggests that both Zeeman and
thermal effects are important to resistively-detected NMR in this regime.Comment: 5 pages, 4 figures. Version accepted for publication in Phys. Rev.B,
Rapid Communication
Surface Acoustic Wave Propagation and Inhomogeneities in Low Density Two-Dimensional Electron Systems Near the Metal-Insulator Transition
We have measured the surface acoustic wave velocity shift in a GaAs/AlGaAs
heterostructure containing a two-dimensional electron system (2DES) in a
low-density regime ( cm) at zero magnetic field. The
interaction of the surface acoustic wave with the 2DES is not well described by
a simple model using low-frequency conductivity measurements. We speculate that
this conflict is a result of inhomogeneities in the 2DES which become very
important at low density. This has implications for the putative
metal-insulator transition in two dimensions.Comment: 5 pages, 5 figure
Growth models, random matrices and Painleve transcendents
The Hammersley process relates to the statistical properties of the maximum
length of all up/right paths connecting random points of a given density in the
unit square from (0,0) to (1,1). This process can also be interpreted in terms
of the height of the polynuclear growth model, or the length of the longest
increasing subsequence in a random permutation. The cumulative distribution of
the longest path length can be written in terms of an average over the unitary
group. Versions of the Hammersley process in which the points are constrained
to have certain symmetries of the square allow similar formulas. The derivation
of these formulas is reviewed. Generalizing the original model to have point
sources along two boundaries of the square, and appropriately scaling the
parameters gives a model in the KPZ universality class. Following works of Baik
and Rains, and Pr\"ahofer and Spohn, we review the calculation of the scaled
cumulative distribution, in which a particular Painlev\'e II transcendent plays
a prominent role.Comment: 27 pages, 5 figure
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
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