Geometric decay of infection probabilities for the anisotropic contact process

Abstract

Consider the anisotropic symmetric contact process on a homogeneous tree T 2d of degree 2 d⩾2 with a single initially infected site at the root vertex of the tree. We show that, for all values of the infection vector λ, each integer n⩾1, and each vertex x∈ T 2d at distance n from the root vertex, the probability P (x is ever infected )=u x(λ) satisfies u x ( λ)⩽[ β c ( λ)] n−1 for some function β c that we will specify. This geometric decay property governs the growth and dispersal behaviour of the process and lies at the core of the method of Hueter (preprint, arXiv: math.PR/0109047), which applies the thermodynamic formalism and the theory of Gibbs states by Bowen (Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer, Berlin, 1975) to the contact process on trees. We leave open the question as to when (if at all) λ c is the maximal infection rate among the components of λ.