Nonintersecting Brownian bridges on the unit circle with drift

Abstract

Nonintersecting Brownian bridges on the unit circle form a determinantal stochastic process exhibiting random matrix statistics for large numbers of walkers. We investigate the effect of adding a drift term μ to walkers on the circle conditioned to start and end at the same position. For each return time T<π2 we show there exists a critical drift μc(T) such that if |μ−2πm/T|<μc(T) for some integer m then the expected winding number for each walker is asymptotically m. In addition, we compute the asymptotic distribution of total winding numbers in the double-scaling regime in which the expected number of walkers with winding number not equal to m is finite. The method of proof is Riemann–Hilbert analysis of a certain family of discrete orthogonal polynomials with varying complex exponential weights. This is the first asymptotic analysis of such a class of polynomials. We determine asymptotic formulas for these polynomials as the degree of the polynomial grows large and demonstrate the emergence of a second band of zeros by a mechanism not previously seen for discrete orthogonal polynomials with real weights.