  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.
