An Improved Lower Bound for the Critical Parameter of the Stavskaya's Process via a Generalized Recurrent Method

TL;DR

This paper improves the lower bound for the critical parameter of Stavskaya's process by extending a recurrent method with increased memory, leading to a more precise estimate of the phase transition point.

Contribution

The authors generalize and extend a recurrent method by increasing the walk's memory and forbidden sequences, achieving a tighter lower bound for the critical parameter.

Findings

New lower bound: .1370721 for      

Extended method with up to 20-step memory improves previous bounds

Numerical optimization of spectral radius under the new framework

Abstract

Stavskaya's process, a discrete-time version of the contact process on $\Z$, is known to exhibit a phase transition at a critical parameter $\alpc$ whose exact value remains an open problem. Recent work by Ramos et al. established a lower bound by linking the process's survival to the non-percolation of a dual contour. The probability of this contour was estimated using a recurrent method on a state space of weighted random walks with short-term memory. In this paper, we generalize and extend this method by systematically increasing the walk's memory and enriching the set of forbidden path sequences. By increasing the memory up to a length of 20 steps (corresponding to our parameter $n=7$), we formulate the problem with a one-step transition matrix and numerically optimize its spectral radius. We thus establish the new lower bound $\alpc > 0.1370721$.