  We revisit the classical problem of diffusion of a scalar (or heat) released
in a two-dimensional medium with an embedded periodic array of impermeable
obstacles such as perforations. Homogenisation theory provides a coarse-grained
description of the scalar at large times and predicts that it diffuses with a
certain effective diffusivity, so the concentration is approximately Gaussian.
We improve on this by developing a large-deviation approximation which also
captures the non-Gaussian tails of the concentration through a rate function
obtained by solving a family of eigenvalue problems. We focus on cylindrical
obstacles and on the dense limit, when the obstacles occupy a large area
fraction and non-Gaussianity is most marked. We derive an asymptotic
approximation for the rate function in this limit, valid uniformly over a wide
range of distances. We use finite-element implementations to solve the
eigenvalue problems yielding the rate function for arbitrary obstacle area
fractions and an elliptic boundary-value problem arising in the asymptotics
calculation. Comparison between numerical results and asymptotic predictions
confirm the validity of the latter.
