First occurrence time of a large density fluctuation for a system of independent random walks
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.
scientific article; zbMATH DE number 1501490