Non-intersecting Brownian walkers and Yang-Mills theory on the sphere

TL;DR

This paper links non-intersecting Brownian motions with 2D Yang-Mills theory on a sphere, revealing a third-order phase transition and Tracy-Widom distribution universality in the large N limit across different boundary conditions.

Contribution

It establishes a novel mapping between reunion probabilities of Brownian walkers and Yang-Mills partition functions, uncovering phase transitions and universal distributions in these stochastic systems.

Findings

Third-order phase transition at critical system size L_c(N)~√N.

Reunion probabilities match Tracy-Widom distributions (GUE and GOE) near criticality.

Maximal height distribution of Brownian excursions follows GOE Tracy-Widom law.

Abstract

We study a system of N non-intersecting Brownian motions on a line segment [0,L] with periodic, absorbing and reflecting boundary conditions. We show that the normalized reunion probabilities of these Brownian motions in the three models can be mapped to the partition function of two-dimensional continuum Yang-Mills theory on a sphere respectively with gauge groups U(N), Sp(2N) and SO(2N). Consequently, we show that in each of these Brownian motion models, as one varies the system size L, a third order phase transition occurs at a critical value L=L_c(N)\sim \sqrt{N} in the large N limit. Close to the critical point, the reunion probability, properly centered and scaled, is identical to the Tracy-Widom distribution describing the probability distribution of the largest eigenvalue of a random matrix. For the periodic case we obtain the Tracy-Widom distribution corresponding to the GUE random matrices, while for the absorbing and reflecting cases we get the Tracy-Widom distribution corresponding to GOE random matrices. In the absorbing case, the reunion probability is also identified as the maximal height of N non-intersecting Brownian excursions ("watermelons" with a wall) whose distribution in the asymptotic scaling limit is then described by GOE Tracy-Widom law. In addition, large deviation formulas for the maximum height are also computed.