Contact Process for the Spread of Knowledge

Abstract

This paper is concerned with a natural variant of the contact process modeling the spread of knowledge on the integer lattice. Each site is characterized by its knowledge, measured by a real number ranging from 0 = ignorant to 1 = omniscient. Neighbors interact at rate \(\lambda \), which results in both neighbors attempting to teach each other a fraction \(\mu \) of their knowledge, and individuals die at rate one, which results in a new individual with no knowledge. Starting with a single omniscient site, our objective is to study whether the total amount of knowledge on the lattice converges to zero (extinction) or remains bounded away from zero (survival). The process dies out when \(\lambda \le \lambda _c\) and/or \(\mu = 0\), where \(\lambda _c\) denotes the critical value of the contact process. In contrast, we prove that, for all \(\lambda > \lambda _c\), there is a unique phase transition in the direction of \(\mu \), and for all \(\mu > 0\), there is a unique phase transition in the direction of \(\lambda \). Our proof of survival relies on block constructions showing more generally convergence of the knowledge to infinity, while our proof of extinction relies on martingale techniques showing more generally an exponential decay of the knowledge.