Contact Processes on Random Regular Graphs by Steven Lalley

Contact Processes on Random Regular Graphs

We show that the contact process on a random d-regular graph initiated by a single infected vertex obeys the “cutoff phenomenon” in its supercritical phase. In particular, we prove that, when the infection rate is larger than the lower critical value of the contact process on the infinite d-regular tree, there are positive constants C, p depending on the infection rate such that for any ε > 0, ...

The weak/strong survival transition on trees and nonamenable graphs

Various stochastic processes on nonamenable graphs and manifolds of exponential volume growth exhibit phases that do not occur in the corresponding processes on amenable graphs. Examples include: (1) branching diffusion and random walk on hyperbolic space, which for intermediate branching rates may survive globally but not locally; (2) contact processes on homogeneous trees, which likewise can ...

Random Walks on Regular Languages and Algebraic Systems of Generating Functions

A random walk on a regular language is a Markov chain on the set of all finite words from a finite alphabet A whose transition probabilities obey the following rules: (1) Only the last two letters of a word may be modified in one jump, and at most one letter may be adjoined or deleted. (2) Probabilities of modification, deletion, and/or adjunction depend only on the last two letters of the curr...

Anisotropic Contact Processes on Homogeneous Trees

Sufficient conditions for the existence of a weak survival phase are given for an anisotropic contact process on a homogeneous tree. These require that the contact process be homogeneous, that is, for any two vertices x, y of the tree there is an automorphism mapping x to y leaving the infection rates invariant; and that the contact process be weakly symmetric, that is, for each vertex there sh...

Random Series in Powers of Algebraic Integers