  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.
