The Annals of Probability

Nonintersecting Brownian motions on the unit circle

Karl Liechty and Dong Wang

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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.

Article information

Ann. Probab., Volume 44, Number 2 (2016), 1134-1211.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 35Q15: Riemann-Hilbert problems [See also 30E25, 31A25, 31B20] 42C05: Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45]

Nonintersecting Brownian motions determinantal process discrete orthogonal polynomial tacnode process Pearcey process Riemann–Hilbert problem double contour integral formula


Liechty, Karl; Wang, Dong. Nonintersecting Brownian motions on the unit circle. Ann. Probab. 44 (2016), no. 2, 1134--1211. doi:10.1214/14-AOP998.

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