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March 2016 Nonintersecting Brownian motions on the unit circle
Karl Liechty, Dong Wang
Ann. Probab. 44(2): 1134-1211 (March 2016). DOI: 10.1214/14-AOP998


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.


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Karl Liechty. Dong Wang. "Nonintersecting Brownian motions on the unit circle." Ann. Probab. 44 (2) 1134 - 1211, March 2016.


Received: 1 July 2014; Revised: 1 December 2014; Published: March 2016
First available in Project Euclid: 14 March 2016

zbMATH: 1342.60138
MathSciNet: MR3474469
Digital Object Identifier: 10.1214/14-AOP998

Primary: 60J65
Secondary: 35Q15 , 42C05

Keywords: Determinantal process , discrete orthogonal polynomial , double contour integral formula , Nonintersecting Brownian motions , Pearcey process , Riemann–Hilbert problem , Tacnode process

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 2 • March 2016
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