Nonintersecting Brownian motions on the unit circle

Abstract

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 Painlevé 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.

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)

References in corpus (6)

• The Pearcey Process
• Non-intersecting Brownian walkers and Yang-Mills theory on the sphere
• Exact distribution of the maximal height of p vicious walkers
• Differential Equations for Dyson Processes
• PDEs for the joint distributions of the Dyson, Airy and Sine processes
• Noncolliding Brownian Motion and Determinantal Processes

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• Elliptic Bessel processes and elliptic Dyson models realized as temporally inhomogeneous processes
• Dip-ramp-plateau for Dyson Brownian motion from the identity on $U(N)$
• Critical Behavior of Non-Intersecting Brownian Motions
• Cyclic Pólya Ensembles on the Unitary Matrices and their Spectral Statistics
• Macdonald denominators for affine root systems, orthogonal theta functions, and elliptic determinantal point processes
• Loop-erased walks and random matrices