Reunion Probability of N Vicious Walkers: Typical and Large Fluctuations for Large N

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 (called "watermelons" with a wall) 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 analyze the asymptotic behavior of this reunion probability for large N using two complementary techniques: (i) a saddl..