First occurrence time of a large density fluctuation for a system of independent random walks (Q1577465)

A Poisson distributed number of particles is assigned to each site of the \(d\)-dimensional integer lattice. The mean number of particles is the same at all sites. The assignments are done independently. Then each particle starts a continuous time symmetric nearest neighbors random walk. The initial distribution is stationary for the motion. The authors study the first time \(T_n\) that a hypercube of volume \(n^d\) is occupied by more than \(\rho'n^d\) particles, where \(\rho'>\rho\) (a large deviation of the density of particles in the hypercube). They prove that as \(n\) goes to infinity, the time conveniently normalized converges to an exponentially distributed random variable.