  The dispersion of a diffusive scalar in a fluid flowing through a network has
many applications including to biological flows, porous media, water supply and
urban pollution. Motivated by this, we develop a large-deviation theory that
predicts the evolution of the concentration of a scalar released in a
rectangular network in the limit of large time $t \gg 1$. This theory provides
an approximation for the concentration that remains valid for large distances
from the centre of mass, specifically for distances up to $O(t)$ and thus much
beyond the $O(t^{1/2})$ range where a standard Gaussian approximation holds. A
byproduct of the approach is a closed-form expression for the effective
diffusivity tensor that governs this Gaussian approximation. Monte Carlo
simulations of Brownian particles confirm the large-deviation results and
demonstrate their effectiveness in describing the scalar distribution when $t$
is only moderately large.
