Non-intersecting Brownian Bridges in the Flat-to-Flat Geometry

We study N vicious Brownian bridges propagating from an initial configuration {a(1) < a(2) < ... < a(N)) at time t = 0 to a final configuration {b(1) < b(2) < ... < b(N)) at time t = t(f), while staying non-intersecting for all 0 <= t <= t(f). We first show that this problem can be mapped to a non-intersecting Dyson's Brownian bridges with Dyson index beta = 2. For the latter we derive an exact effective Langevin equation that allows to generate very efficiently the vicious bridge configurations. In particular, for the flat-to-flat configuration in the large N limit, where a(i) = b(i) = (i - 1)/N, for i = 1, ..., N, we use this effective Langevin equation to derive an exact Burgers' equation (in the inviscid limit) for the Green's function and solve this Burgers' equation for arbitrary time 0 <= t <= t(f). At certain specific values of intermediate times t, such as t = t(f)/2, t = t(f)/3 and t = t(f)/4 we obtain the average density of the flat-to-flat bridge explicitly. We also derive explicitly how the two edges of the average density evolve from time t = 0 to time t = t(f). Finally, we discuss connections to some well known problems, such as the Chem-Simons model, the related Stieltjes-Wigert orthogonal polynomials and the Borodin-Muttalib ensemble of determinantal point processes.