  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.
