  When does a diffusing particle reach its target for the first time? This
first-passage time (FPT) problem is central to the kinetics of molecular
reactions in chemistry and molecular biology. Here we explain the behavior of
smooth FPT densities, for which all moments are finite, and demonstrate
universal yet generally non-Poissonian long-time asymptotics for a broad
variety of transport processes. While Poisson-like asymptotics arise
generically in the presence of an effective repulsion in the immediate vicinity
of the target, a time-scale separation between direct and reflected indirect
trajectories gives rise to a universal proximity effect: Direct paths, heading
more or less straight from the point of release to the target, become typical
and focused, with a narrow spread of the corresponding first passage times.
Conversely, statistically dominant indirect paths exploring the system size
tend to be massively dissimilar. The initial distance to the target
particularly impacts gene regulatory or competitive stochastic processes, for
which often few binding events determine the regulatory outcome. The proximity
effect is independent of details of the transport, highlighting the robust
character of the FPT features uncovered here.
