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

We study N vicious Brownian bridges propagating from an initial configuration {a1 < a2 < . . . < aN} at time t = 0 to a final configuration {b1 < b2 < . . . < bN} at time t = tf , while staying non-intersecting for all 0 ≤ t ≤ tf . We first show that this problem can be mapped to a nonintersecting Dyson’s Brownian bridges with Dyson index β = 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 ai = bi = (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 ≤ tf . At certain specific values of intermediate times t, such as t = tf/2, t = tf/3 and t = tf/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 = tf . Finally, we discuss connections to some well known problems, such as the Chern-Simons model, the related Stieltjes-Wigert orthogonal polynomials and the Borodin-Muttalib ensemble of determinantal point processes.