  We consider a particle diffusing outside a compact planar set and investigate
its boundary local time $\ell_t$, i.e., the rescaled number of encounters
between the particle and the boundary up to time $t$. In the case of a disk,
this is also the (rescaled) number of encounters of two diffusing circular
particles in the plane. For that case, we derive explicit integral
representations for the probability density of the boundary local time $\ell_t$
and for the probability density of the first-crossing time of a given threshold
by $\ell_t$. The latter density is shown to exhibit a very slow long-time decay
due to extremely long diffusive excursions between encounters. We briefly
discuss some practical consequences of this behavior for applications in
chemical physics and biology.
