# Distribution of Brownian coincidences

## TL;DR

This paper analyzes the probability distribution of total pairwise coincidence time among multiple independent Brownian particles, using advanced mathematical mappings to derive explicit formulas and asymptotic behaviors.

## Contribution

It introduces a novel approach by mapping the coincidence time problem onto models like Lieb-Liniger, directed polymers, and KPZ, providing explicit formulas and asymptotics.

## Key findings

- Derived closed-form probability distribution for coincidence time.
- Obtained universal large deviation tail independent of geometry.
- Validated analytical results with numerical simulations.

## Abstract

We study the probability distribution, $P_N(T)$, of the coincidence time $T$, i.e. the total local time of all pairwise coincidences of $N$ independent Brownian walkers. We consider in details two geometries: Brownian motions all starting from $0$, and Brownian bridges. Using a Feynman-Kac 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, $P_N(T) \sim \exp(- 3 T^2/(N^3-N))$ 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.