3,481 research outputs found
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
Isomonodromic deformation theory and the next-to-diagonal correlations of the anisotropic square lattice Ising model
In 1980 Jimbo and Miwa evaluated the diagonal two-point correlation function
of the square lattice Ising model as a -function of the sixth Painlev\'e
system by constructing an associated isomonodromic system within their theory
of holonomic quantum fields. More recently an alternative isomonodromy theory
was constructed based on bi-orthogonal polynomials on the unit circle with
regular semi-classical weights, for which the diagonal Ising correlations arise
as the leading coefficient of the polynomials specialised appropriately. Here
we demonstrate that the next-to-diagonal correlations of the anisotropic Ising
model are evaluated as one of the elements of this isomonodromic system or
essentially as the Cauchy-Hilbert transform of one of the bi-orthogonal
polynomials.Comment: 11 pages, 1 figur
Boundary conditions associated with the Painlev\'e III' and V evaluations of some random matrix averages
In a previous work a random matrix average for the Laguerre unitary ensemble,
generalising the generating function for the probability that an interval at the hard edge contains eigenvalues, was evaluated in terms of
a Painlev\'e V transcendent in -form. However the boundary conditions
for the corresponding differential equation were not specified for the full
parameter space. Here this task is accomplished in general, and the obtained
functional form is compared against the most general small behaviour of
the Painlev\'e V equation in -form known from the work of Jimbo. An
analogous study is carried out for the the hard edge scaling limit of the
random matrix average, which we have previously evaluated in terms of a
Painlev\'e \IIId transcendent in -form. An application of the latter
result is given to the rapid evaluation of a Hankel determinant appearing in a
recent work of Conrey, Rubinstein and Snaith relating to the derivative of the
Riemann zeta function
Governance factors affecting the ecological performance of Marine Reserves in the wider Caribbean
Semi-classical Laguerre polynomials and a third order discrete integrable equation
A semi-discrete Lax pair formed from the differential system and recurrence
relation for semi-classical orthogonal polynomials, leads to a discrete
integrable equation for a specific semi-classical orthogonal polynomial weight.
The main example we use is a semi-classical Laguerre weight to derive a third
order difference equation with a corresponding Lax pair.Comment: 11 page
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
Scaling limit of vicious walks and two-matrix model
We consider the diffusion scaling limit of the one-dimensional vicious walker
model of Fisher and derive a system of nonintersecting Brownian motions. The
spatial distribution of particles is studied and it is described by use of
the probability density function of eigenvalues of Gaussian random
matrices. The particle distribution depends on the ratio of the observation
time and the time interval in which the nonintersecting condition is
imposed. As is going on from 0 to 1, there occurs a transition of
distribution, which is identified with the transition observed in the
two-matrix model of Pandey and Mehta. Despite of the absence of matrix
structure in the original vicious walker model, in the diffusion scaling limit,
accumulation of contact repulsive interactions realizes the correlated
distribution of eigenvalues in the multimatrix model as the particle
distribution.Comment: REVTeX4, 12 pages, no figure, minor corrections made for publicatio
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
Moments of vicious walkers and M\"obius graph expansions
A system of Brownian motions in one-dimension all started from the origin and
conditioned never to collide with each other in a given finite time-interval
is studied. The spatial distribution of such vicious walkers can be
described by using the repulsive eigenvalue-statistics of random Hermitian
matrices and it was shown that the present vicious walker model exhibits a
transition from the Gaussian unitary ensemble (GUE) statistics to the Gaussian
orthogonal ensemble (GOE) statistics as the time is going on from 0 to .
In the present paper, we characterize this GUE-to-GOE transition by presenting
the graphical expansion formula for the moments of positions of vicious
walkers. In the GUE limit , only the ribbon graphs contribute and the
problem is reduced to the classification of orientable surfaces by genus.
Following the time evolution of the vicious walkers, however, the graphs with
twisted ribbons, called M\"obius graphs, increase their contribution to our
expansion formula, and we have to deal with the topology of non-orientable
surfaces. Application of the recent exact result of dynamical correlation
functions yields closed expressions for the coefficients in the M\"obius
expansion using the Stirling numbers of the first kind.Comment: REVTeX4, 11 pages, 1 figure. v.2: calculations of the Green function
and references added. v.3: minor additions and corrections made for
publication in Phys.Rev.
Eigenvalue Separation in Some Random Matrix Models
The eigenvalue density for members of the Gaussian orthogonal and unitary
ensembles follows the Wigner semi-circle law. If the Gaussian entries are all
shifted by a constant amount c/Sqrt(2N), where N is the size of the matrix, in
the large N limit a single eigenvalue will separate from the support of the
Wigner semi-circle provided c > 1. In this study, using an asymptotic analysis
of the secular equation for the eigenvalue condition, we compare this effect to
analogous effects occurring in general variance Wishart matrices and matrices
from the shifted mean chiral ensemble. We undertake an analogous comparative
study of eigenvalue separation properties when the size of the matrices are
fixed and c goes to infinity, and higher rank analogues of this setting. This
is done using exact expressions for eigenvalue probability densities in terms
of generalized hypergeometric functions, and using the interpretation of the
latter as a Green function in the Dyson Brownian motion model. For the shifted
mean Gaussian unitary ensemble and its analogues an alternative approach is to
use exact expressions for the correlation functions in terms of classical
orthogonal polynomials and associated multiple generalizations. By using these
exact expressions to compute and plot the eigenvalue density, illustrations of
the various eigenvalue separation effects are obtained.Comment: 25 pages, 9 figures include
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