Growth rates in the branching random walk

Abstract

We will consider the branching random walk on the real line. An initial ancestor is at the origin. He has children, the first generation, and these have positions which form a point process on the line. These children in their turn have children, independently of each other. The positions of the children of a first generation person, relative to his own position form a point process; this point process has the same distributions as the one giving the positions of the initial ancestor's children. This gives the second generation. Subsequent generations are formed in the same way. Branching random walks are essentially the same as spatially homogeneous branching processes and are closely related to cluster fields, both of which are discussed in [14]. Let {z~ ")} be the positions of the nth generation people and let