  We study the long time behavior of bounded, integrable solutions to a
nonlocal diffusion equation, $\partial _t u=J*u-u$, where $J$ is a smooth,
radially symmetric kernel with support $B_d(0)\subset\mathbb{R}^2$. The problem
is set in an exterior two-dimensional domain which excludes a hole
$\mathcal{H}$, and with zero Dirichlet data on $\mathcal{H}$. In the far field
scale, $\xi_1\le |x|t^{-1/2}\le \xi_2$ with $\xi_1,\xi_2>0$, the scaled
function $\log t\, u(x,t)$ behaves as a multiple of the fundamental solution
for the local heat equation with a certain diffusivity determined by $J$. The
proportionality constant, which characterizes the first non-trivial term in the
asymptotic behavior of the mass, is given by means of the asymptotic \lq
logarithmic momentum' of the solution,
$\lim_{t\to\infty}\int_{\mathbb{R}^2}u(x,t)\log|x|\,dx$. This asymptotic
quantity can be easily computed in terms of the initial data. In the near field
scale, $|x|\le t^{1/2}h(t)$ with $\lim_{t\to\infty} h(t)=0$, the scaled
function $t(\log t)^2u(x,t)/\log |x|$ converges to a multiple of $\phi(x)/\log
|x|$, where $\phi$ is the unique stationary solution of the problem that
behaves as $\log|x|$ when $|x|\to\infty$. The proportionality constant is
obtained through a matching procedure with the far field limit. Finally, in the
very far field, $|x|\ge t^{1/2} g(t)$ with $g(t)\to\infty$, the solution is
proved to be of order $o((t\log t)^{-1})$.
