45 research outputs found
Exact solution of the six-vertex model with domain wall boundary condition. Critical line between ferroelectric and disordered phases
This is a continuation of the papers [4] of Bleher and Fokin and [5] of
Bleher and Liechty, in which the large asymptotics is obtained for the
partition function of the six-vertex model with domain wall boundary
conditions in the disordered and ferroelectric phases, respectively. In the
present paper we obtain the large asymptotics of on the critical line
between these two phases.Comment: 22 pages, 6 figures, to appear in the Journal of Statistical Physic
Large n limit of Gaussian random matrices with external source, Part III: Double scaling limit
We consider the double scaling limit in the random matrix ensemble with an
external source \frac{1}{Z_n} e^{-n \Tr({1/2}M^2 -AM)} dM defined on Hermitian matrices, where is a diagonal matrix with two eigenvalues of equal multiplicities. The value is critical since the eigenvalues
of accumulate as on two intervals for and on one
interval for . These two cases were treated in Parts I and II, where
we showed that the local eigenvalue correlations have the universal limiting
behavior known from unitary random matrix ensembles. For the critical case
new limiting behavior occurs which is described in terms of Pearcey
integrals, as shown by Br\'ezin and Hikami, and Tracy and Widom. We establish
this result by applying the Deift/Zhou steepest descent method to a -matrix valued Riemann-Hilbert problem which involves the construction of a
local parametrix out of Pearcey integrals. We resolve the main technical issue
of matching the local Pearcey parametrix with a global outside parametrix by
modifying an underlying Riemann surface.Comment: 36 pages, 9 figure
Unique positive solution for an alternative discrete PainlevĂŠ I equation
We show that the alternative discrete Painleve I equation has a unique solution which remains positive for all n >0. Furthermore, we identify this positive solution in terms of a special solution of the second Painleve equation involving the Airy function Ai(t). The special-function solutions of the second Painleve equation involving only the Airy function Ai(t) therefore have the property that they remain positive for all n>0 and all t>0, which is a new characterization of these special solutions of the second PainlevĂŠ equation and the alternative discrete PainlevĂŠ I equation
Nearest Neighbor Distances on a Circle: Multidimensional Case
We study the distances, called spacings, between pairs of neighboring energy
levels for the quantum harmonic oscillator. Specifically, we consider all
energy levels falling between E and E+1, and study how the spacings between
these levels change for various choices of E, particularly when E goes to
infinity. Primarily, we study the case in which the spring constant is a badly
approximable vector. We first give the proof by Boshernitzan-Dyson that the
number of distinct spacings has a uniform bound independent of E. Then, if the
spring constant has components forming a basis of an algebraic number field, we
show that, when normalized up to a unit, the spacings are from a finite set.
Moreover, in the specific case that the field has one fundamental unit, the
probability distribution of these spacings behaves quasiperiodically in log E.
We conclude by studying the spacings in the case that the spring constant is
not badly approximable, providing examples for which the number of distinct
spacings is unbounded.Comment: Version 2 is updated to include more discussion of previous works. 17
pages with five figures. To appear in the Journal of Statistical Physic
Level spacing statistics of classically integrable systems -Investigation along the line of the Berry-Robnik approach-
By extending the approach of Berry and Robnik, the limiting level spacing
distribution of a system consisting of infinitely many independent components
is investigated. The limiting level spacing distribution is characterized by a
single monotonically increasing function of the level spacing
. Three cases are distinguished: (i) Poissonian if ,
(ii) Poissonian for large , but possibly not for small if
, and (iii) sub-Poissonian if .
This implies that, even when energy-level distributions of individual
components are statistically independent, non-Poissonian level spacing
distributions are possible.Comment: 19 pages, 4 figures. Accepted for publication in Phys. Rev.
Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights
We study a model of non-intersecting squared Bessel processes in the
confluent case: all paths start at time at the same positive value , remain positive, and are conditioned to end at time at . In
the limit , after appropriate rescaling, the paths fill out a
region in the -plane that we describe explicitly. In particular, the paths
initially stay away from the hard edge at , but at a certain critical
time the smallest paths hit the hard edge and from then on are stuck to
it. For we obtain the usual scaling limits from random matrix
theory, namely the sine, Airy, and Bessel kernels. A key fact is that the
positions of the paths at any time constitute a multiple orthogonal
polynomial ensemble, corresponding to a system of two modified Bessel-type
weights. As a consequence, there is a matrix valued
Riemann-Hilbert problem characterizing this model, that we analyze in the large
limit using the Deift-Zhou steepest descent method. There are some novel
ingredients in the Riemann-Hilbert analysis that are of independent interest.Comment: 59 pages, 11 figure
Reconstruction of Random Colourings
Reconstruction problems have been studied in a number of contexts including
biology, information theory and and statistical physics. We consider the
reconstruction problem for random -colourings on the -ary tree for
large . Bhatnagar et. al. showed non-reconstruction when and reconstruction when . We tighten this result and show non-reconstruction when and reconstruction when .Comment: Added references, updated notatio
Decay of the Sinai Well in D dimensions
We study the decay law of the Sinai Well in dimensions and relate the
behavior of the decay law to internal distributions that characterize the
dynamics of the system. We show that the long time tail of the decay is
algebraic (), irrespective of the dimension .Comment: 14 pages, Figures available under request. Revtex. Submitted to Phys.
Rev. E.,e-mail: [email protected]
A Contour Method on Cayley tree
We consider a finite range lattice models on Cayley tree with two basic
properties: the existence of only a finite number of ground states and with
Peierls type condition. We define notion of a contour for the model on the
Cayley tree. By a contour argument we show the existence of different
(where is the number of ground states) Gibbs measures.Comment: 12 page
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