Nonintersecting Brownian bridges on the unit circle with drift

TL;DR

This paper analyzes how adding a drift affects the winding number distribution of nonintersecting Brownian bridges on the circle, revealing phase transitions and novel zero distributions through advanced Riemann-Hilbert techniques.

Contribution

It introduces the first asymptotic analysis of a new class of discrete orthogonal polynomials with complex weights, uncovering unique zero band phenomena.

Findings

Expected total winding number is asymptotically zero below a critical drift.

Derived asymptotic distribution of winding numbers in the double-scaling regime.

Identified emergence of a second zero band mechanism not seen before.

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 to walkers on the circle conditioned to start and end at the same position. For each return time $T<\pi^2$ we show that if the absolute value of the drift is less than a critical value then the expected total winding number is asymptotically zero. In addition, we compute the asymptotic distribution of total winding numbers in the double-scaling regime in which the expected total winding 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 and demonstrate the emergence of a second band of zeros by a mechanism not previously seen for discrete orthogonal polynomials with real weights.