12 research outputs found
Extremal statistics of curved growing interfaces in 1+1 dimensions
We study the joint probability distribution function (pdf) of the maximum M
of the height and its position X_M of a curved growing interface belonging to
the universality class described by the Kardar-Parisi-Zhang equation in 1+1
dimensions. We obtain exact results for the closely related problem of p
non-intersecting Brownian bridges where we compute the joint pdf P_p(M,\tau_M)
where \tau_M is there the time at which the maximal height M is reached. Our
analytical results, in the limit p \to \infty, become exact for the interface
problem in the growth regime. We show that our results, for moderate values of
p \sim 10 describe accurately our numerical data of a prototype of these
systems, the polynuclear growth model in droplet geometry. We also discuss
applications of our results to the ground state configuration of the directed
polymer in a random potential with one fixed endpoint.Comment: 6 pages, 4 figures. Published version, to appear in Europhysics
Letters. New results added for non-intersecting excursion
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
Endpoint distribution of directed polymers in 1+1 dimensions
We give an explicit formula for the joint density of the max and argmax of
the Airy process minus a parabola. The argmax has a universal distribution
which governs the rescaled endpoint for large time or temperature of directed
polymers in 1+1 dimensions.Comment: Expanded introductio
Determinantal process starting from an orthogonal symmetry is a Pfaffian process
When the number of particles is finite, the noncolliding Brownian motion
(BM) and the noncolliding squared Bessel process with index
(BESQ) are determinantal processes for arbitrary fixed initial
configurations. In the present paper we prove that, if initial configurations
are distributed with orthogonal symmetry, they are Pfaffian processes in the
sense that any multitime correlation functions are expressed by Pfaffians. The
skew-symmetric matrix-valued correlation kernels of the Pfaffians
processes are explicitly obtained by the equivalence between the noncolliding
BM and an appropriate dilatation of a time reversal of the temporally
inhomogeneous version of noncolliding BM with finite duration in which all
particles start from the origin, , and by the equivalence between
the noncolliding BESQ and that of the noncolliding squared
generalized meander starting from .Comment: v2: AMS-LaTeX, 17 pages, no figure, corrections made for publication
in J.Stat.Phy
Airy processes and variational problems
We review the Airy processes; their formulation and how they are conjectured
to govern the large time, large distance spatial fluctuations of one
dimensional random growth models. We also describe formulas which express the
probabilities that they lie below a given curve as Fredholm determinants of
certain boundary value operators, and the several applications of these
formulas to variational problems involving Airy processes that arise in
physical problems, as well as to their local behaviour.Comment: Minor corrections. 41 pages, 4 figures. To appear as chapter in "PASI
Proceedings: Topics in percolative and disordered systems
Extreme value statistics from the Real Space Renormalization Group: Brownian Motion, Bessel Processes and Continuous Time Random Walks
We use the Real Space Renormalization Group (RSRG) method to study extreme
value statistics for a variety of Brownian motions, free or constrained such as
the Brownian bridge, excursion, meander and reflected bridge, recovering some
standard results, and extending others. We apply the same method to compute the
distribution of extrema of Bessel processes. We briefly show how the continuous
time random walk (CTRW) corresponds to a non standard fixed point of the RSRG
transformation.Comment: 24 pages, 5 figure
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