  We study the time until first occurrence, the first-passage time, of rare
density fluctuations in diffusive systems. We approach the problem using a
model consisting of many independent random walkers on a lattice. The existence
of spatial correlations makes this problem analytically intractable. However,
for a mean-field approximation in which the walkers can jump anywhere in the
system, we obtain a simple asymptotic form for the mean first-passage time to
have a given number k of particles at a distinguished site. We show
numerically, and argue heuristically, that for large enough k, the mean-field
results give a good approximation for first-passage times for systems with
nearest-neighbour dynamics, especially for two and higher spatial dimensions.
Finally, we show how the results change when density fluctuations anywhere in
the system, rather than at a specific distinguished site, are considered.
