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