Lower and Upper Bounds for the Survival of Infinite Absorbing Markov Chains

Abstract

A wide variety of problems in the domain of directed percolation theory, con- tact process, voter models rely on the analysis of a infinite absorbing Markov chain. Although results for these problems have been obtained through other approaches, reasoning about the Markov chain gives rise to elegant results that are often explicit functions of the chain parameters. This allows results to be re- used across a wide variety of problems. In this paper, we present results for the survival of a class of discrete time Markov processes whose states are finite sets of integers. We present lower bounds using two different approaches as well as an upper bound. We borrow varied techniques and show in detail how they can be used to analyse Markov chains of this nature. The results presented in this paper can be used to derive many results in percolation theory in a completely independent manner.