Non-intersecting Brownian bridges in the flat-to-flat geometry

We study $N$ vicious Brownian bridges propagating from an initial configuration $\{a_1<a_2<\dots <a_N\}$ at time $t=0$ to a final configuration $\{b_1<b_2<\dots <b_N\}$ 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 $a_i=b_i=(i-1)\mathrm{/}N$, for $i=1,\cdots ,N$, 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 $0 \leq t\leq t_f$\mathrm{.}Atcertainspecificvaluesofintermediatetimes$t$,suchas$t=t_f/2$,$t=t_f/3$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.