Asymptotic Shape for the Branching Exclusion Process

1998, v.4, №4, 535-547

ABSTRACT

We consider an interacting particle system on $Z^d$ known as branching exclusion process. In this model particles jump onto the nearest neighbor sites at rate ${\lambda / 2}$ but can also create an offspring at rate ${1/2d}.$ Any creation or jump that violates the at-most-one-particle-per-site rule does not take effect. The process begins with one particle at the origin. Durrett and Griffeath [Contact process in several dimensions, Z. Wahrsch. verw. Geb., 1982, 59, 535-552] proved that the contour of the set of occupied sites at time $t$ rescaled by $t,$ converges almost surely to $A(\lambda),$ a convex set, non-empty in $R^d.$ Here we prove that $A(\lambda)$ rescaled by ${\sqrt \lambda},$ converges to an Euclidean ball.

Keywords: asymptotic shape,branching exclusion process,comparison theorem