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.

我们研究了N 个恶性布朗桥从初始配置\(\{a_1< a_2< \ldots < a_N \}\)在时间\(t=0\)传播到最终配置\(\{b_1< b_2< \ldots < b_N \}\)在时间\(t=t_f\)，同时对于所有\(0\le t \le t_f\)保持不相交。我们首先证明这个问题可以映射到一个不相交的戴森布朗桥，戴森指数\(\beta =2\)。对于后者，我们推导出一个精确有效的朗之万方程，可以非常有效地生成恶性桥梁配置。特别是对于大N限制中的 flat-to-flat 配置，其中\(a_i = b_i = (i-1)/N\)，对于\(i = 1, \ldots , N\)，我们使用这个有效的 Langevin 方程来推导出格林函数的精确 Burgers 方程（在无粘性极限中），并在任意时间求解这个 Burgers 方程\(0 \le t\le t_f\)。在中间时间t 的某些特定值，例如\(t=t_f/2\)、\(t=t_f/3\)和\(t=t_f/4\)我们获得了平面到- 明确地平桥。我们还明确推导出平均密度的两条边如何从时间\(t=0\)到时间\(t=t_f\). 最后，我们讨论了与一些众所周知的问题的联系，例如 Chern-Simons 模型、相关的 Stieltjes-Wigert 正交多项式和行列式点过程的 Borodin-Muttalib 系综。