Critical branching random walks and spatial epidemics.

In the first part we consider a critical nearest neighbor branching random walk on the d-dimensional integer lattice. Denote by V(m) the maximal number of particles at a single site at time m, and by G(m) the event that the branching random walk survives to generation m. We show that if the offspring distribution has finite nth moment, then in dimensions higher than 3, conditional on G( m), V(m) = Op( m1/n); and if the offspring distribution has an exponentially decaying tail, then, conditional on G(m), (a) V( m) = Op(log m) in dimensions higher than 3, and (b) V(m) = Op((log m)2) in dimension 2. On the other hand, we show that if the offspring distribution is non-degenerate, then there exists delta > 0 such that the probability that V( m) is greater than deltalog m goes to 1. Furthermore, we show that, conditional on G(m), in dimensions higher than 3, the number of multiplicity-j sites, j ≥ 1, and the number of occupied sites, normalized by m, converge jointly to multiples of an exponential random variable; in dimension d = 2, however, the number of particles on a "typical" site is Op(log m), and the number of occupied sites is Op( m/log m). In the second part we study spatial SIR epidemics on the d-dimensional integer lattice (d = 2, 3) with village size N. We show that the measure-valued processes associated with these epidemics, under suitable scaling, as N goes to infinity, converge to a standard Dawson-Watanabe process or a Dawson-Watanabe process with killing, depending on whether the initial number of infections is below a critical threshold or at the threshold. A key step in the proof is to show that the local time processes associated with branching random walks, under suitable scaling, converge to the local time density process associated with the limiting super-Brownian motion, establishing a conjecture due to Adler (1993).