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

Abstract

We study $N$ vicious Brownian bridges propagating from an initial configuration $\{a_1<a_2<\dots <a_N\}$ 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 $a_i=b_i=(i-1)/N$, 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 $0\le t\le t_f$. 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.

27 pages, 8 figures

References in corpus (9)

• Non-intersecting Brownian walkers and Yang-Mills theory on the sphere
• Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
• Exact distribution of the maximal height of p vicious walkers
• Invariant $\beta$-ensembles and the Gauss-Wigner crossover
• Nonintersecting Brownian excursions
• Chern-Simons matrix models and Stieltjes-Wigert polynomials
• Tilted elastic lines with columnar and point disorder, non-Hermitian quantum mechanics and spiked random matrices: pinning and localization
• Noncolliding Brownian Motion with Drift and Time-Dependent Stieltjes-Wigert Determinantal Point Process
• Eikonal formulation of large dynamical random matrix models

Cited by in corpus (17)

• Reinforcement learning of rare diffusive dynamics
• Full counting statistics for interacting trapped fermions
• Generating discrete-time constrained random walks and Lévy flights
• Conditioned diffusion processes with an absorbing boundary condition for finite or infinite horizon
• Stability of large complex systems with heterogeneous relaxation dynamics
• Dynamical phase transition in the occupation fraction statistics for non-crossing Brownian particles
• Generating stochastic trajectories with global dynamical constraints
• Generating constrained run-and-tumble trajectories
• Conditioning diffusion processes with killing rates
• Microcanonical conditioning of Markov processes on time-additive observables
• Conditioning diffusion processes with respect to the local time at the origin
• Joint distribution of two Local Times for diffusion processes with the application to the construction of various conditioned processes
• Transport properties of diffusive particles conditioned to survive in trapping environments
• Universality for random matrices with equi-spaced external source: a case study of a biorthogonal ensemble
• Conditioning two diffusion processes with respect to their first-encounter properties
• A Two-HCIZ Gaussian Matrix Model for Non-intersecting Brownian Bridges
• Sampling rare trajectories using stochastic bridges