Geometric decay of infection probabilities for the anisotropic contact process,
Statistics & Probability Letters, 63, 367-374 (2003)

Consider the anisotropic symmetric contact process on a homogeneous tree T_2d of degree 2d > 1 with a single initially infected site at the root vertex of the tree. We show that, for all values of the infection vector \lambda, each integer n > 0, and each vertex x in T_2d at distance n from the root vertex, the probability P ( x is ever infected ) = u_x(\lambda) satisfies u_x(\lambda) \leq ß_c(\lambda)^n-1 for some function ß_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 (1998), which applies the thermodynamic formalism and the theory of Gibbs states (R. Bowen, Springer Lecture Notes in Math., Vol 470, 1975) to the contact process on trees. We leave open the question as to when (if at all) \lambda_c is the maximal infection rate among the components of \lambda.