  The present work investigates the asymptotic behaviors, at the zero-noise
limit, of the first collision-time and first collision-location related to a
pair of self-stabilizing diffusions and of their related particle
approximations. These asymptotic are considered in a peculiar framework where
diffusions evolve in a double-wells landscape and where collisions manifest due
to the combined action of the Brownian motions driving each diffusion and the
action of a self-stabilizing kernel. As the Brownian effects vanish, we show
that first collision-times grow at an explicit exponential rate and that the
related collision-locations persist at a special point in space. These results
are mainly obtained by linking collision phenomena for diffusion processes with
exit-time problems of random perturbed dynamical systems, and by exploiting
Freidlin-Wentzell's LDP approach to solve these exit-time problems.
Importantly, we consider two distinctive situations: the one-dimensional case
(where true collisions can be directly studied) and the general
multidimensional case (where collisions are required to be enlarged).
