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

Abstract

We study N vicious Brownian bridges propagating from an initial configuration \(\{a_1< a_2< \ldots < a_N \}\) at time \(t=0\) to a final configuration \(\{b_1< b_2< \ldots < b_N \}\) at time \(t=t_f\), while staying non-intersecting for all \(0\le t \le t_f\). 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 \(i = 1, \ldots , N\), 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 \(0 \le t\le t_f\). 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 Chern–Simons model, the related Stieltjes–Wigert orthogonal polynomials and the Borodin–Muttalib ensemble of determinantal point processes.