Structure of the Population of Particles for a Branching Random Walk in a Homogeneous Environment

We consider a symmetric branching random walk in a multi-dimensional lattice with continuous time and Markov branching process at each lattice point. It is assumed that initially at each lattice point there is one particle and in the process of branching any particle can produce an arbitrary number of descendants. For a critical process, under the assumption that the walk is transient, we prove the convergence of the distribution of the particle field to the limit stationary distribution. We show the absence of intermittency in the zone $|x-y| = O(\sqrt {t})$, where $x$ and $y$ are spatial coordinates and $t$ is the time, under the assumption of superexponentially light tails of a random walk and a supercriticality of the branching process at the points of the lattice.