Reunion probability of N vicious walkers: typical and large fluctuations for large N

TL;DR

This paper analyzes the reunion probability of N non-intersecting Brownian motions under various boundary conditions, revealing a third-order phase transition at a critical system size and exploring large deviation behaviors with connections to random matrix theory.

Contribution

It provides exact formulas for reunion probabilities of N vicious walkers and characterizes their asymptotic large N behavior, including phase transitions and large deviations, using novel analytical techniques.

Findings

Reunion probability exhibits a third-order phase transition at critical size L_c(N)~√N.

Large deviations in reunion probability are characterized for both tails, revealing unconventional matching regimes.

Connections to Tracy-Widom distributions and the Douglas-Kazakov transition are established.

Abstract

We consider three different models of N non-intersecting Brownian motions on a line segment [0,L] with absorbing (model A), periodic (model B) and reflecting (model C) boundary conditions. In these three cases we study a properly normalized reunion probability, which, in model A, can also be interpreted as the maximal height of N non-intersecting Brownian excursions on the unit time interval. We provide a detailed derivation of the exact formula for these reunion probabilities for finite N using a Fermionic path integral technique. We then analyse the asymptotic behavior of this reunion probability for large N using two complementary techniques: (i) a saddle point analysis of the underlying Coulomb gas and (ii) orthogonal polynomial method. These two methods are complementary in the sense that they work in two different regimes, respectively for L\ll O(\sqrt{N}) and L\geq O(\sqrt{N}). A striking feature of the large N limit of the reunion probability in the three models is that it exhibits a third-order phase transition when the system size L crosses a critical value L=L_c(N)\sim \sqrt{N}. This transition is akin to the Douglas-Kazakov transition in two-dimensional continuum Yang-Mills theory. While the central part of the reunion probability, for L \sim L_c(N), is described in terms of the Tracy-Widom distributions (associated to GOE and GUE depending on the model), the emphasis of the present study is on the large deviations of these reunion probabilities, both in the right [L \gg L_c(N)] and the left [L \ll L_c(N)] tails. In particular, for model B, we find that the matching between the different regimes corresponding to typical L \sim L_c(N) and atypical fluctuations in the right tail L \gg L_c(N) is rather unconventional, compared to the usual behavior found for the distribution of the largest eigenvalue of GUE random matrices.