Distribution of Brownian Coincidences

We study the probability distribution, $P_N\left(T\right)$, of the coincidence time T, i.e. the total time spent by N independent one-dimensional Brownian walkers in a vicinity of vanishing length. We consider in details two geometries: Brownian motions all starting from 0, and Brownian bridges. Using a Feynman–Kač representation for the moment generating function of this coincidence time, we map this problem onto some observables in three related models (i) the propagator of the Lieb–Liniger model of quantum particles with pairwise delta function interactions (ii) the moments of the partition function of a directed polymer in a random medium (iii) the exponential moments of the solution of the Kardar–Parisi–Zhang equation. Using these mappings, we obtain closed formulae for the probability distribution of the coincidence time, its tails and some of its moments. Its asymptotics at large and small coincidence time are also obtained for arbitrary fixed endpoints. The universal large T tail, $P_N\left(T\right)\sim \exp (-3T^2/(N^3-N\left)\right)$ is obtained, and is independent of the geometry. We investigate the large deviations in the limit of a large number of walkers through a Coulomb gas approach. Some of our analytical results are compared with numerical simulations.