We consider an ensemble of $n$ nonintersecting Brownian particles on the unit
circle with diffusion parameter $n^{-1/2}$, which are conditioned to begin at
the same point and to return to that point after time $T$, but otherwise not to
intersect. There is a critical value of $T$ which separates the subcritical
case, in which it is vanishingly unlikely that the particles wrap around the
circle, and the supercritical case, in which particles may wrap around the
circle. In this paper, we show that in the subcritical and critical cases the
probability that the total winding number is zero is almost surely 1 as
$n\to\infty$, and in the supercritical case that the distribution of the total
winding number converges to the discrete normal distribution. We also give a
streamlined approach to identifying the Pearcey and tacnode processes in
scaling limits. The formula of the tacnode correlation kernel is new and
involves a solution to a Lax system for the Painlev\'{e} II equation of size 2
$\times$ 2. The proofs are based on the determinantal structure of the
ensemble, asymptotic results for the related system of discrete Gaussian
orthogonal polynomials, and a formulation of the correlation kernel in terms of
a double contour integral.Comment: Published at http://dx.doi.org/10.1214/14-AOP998 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
We study the distribution of the maximal height of the outermost path in the model of N nonintersecting Brownian motions on the half-line as N → ∞, 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 Airy 2 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.
Abstract. The maximal point of the Airy 2 process minus a parabola is believed to describe the scaling limit of the end-point of the directed polymer in a random medium. This was proved to be true for a few specific cases. Recently two different formulas for the joint distribution of the location and the height of this maximal point were obtained, one by Moreno Flores, Quastel and Remenik, and the other by Schehr. The first formula is given in terms of the Airy function and an associated operator, and the second formula is expressed in terms of the Lax pair equations of the Painlevé II equation. We give a direct proof that these two formulas are the same.
This is a continuation of the paper [4] of Bleher and Fokin, in which the large n asymptotics is obtained for the partition function Z n of the six-vertex model with domain wall boundary conditions in the disordered phase. In the present paper we obtain the large n asymptotics of Z n in the ferroelectric phase. We prove that for any ε > 0, as n → ∞, Z n = CG n F n 2 [1 + O(e −n 1−ε )], and we find the exact values of the constants C, G and F . The proof is based on the large n asymptotics for the underlying discrete orthogonal polynomials and on the Toda equation for the tau-function.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.