The Power Light Cone of the Discrete Bak-Sneppen, Contact and other local processes

Abstract

We consider a class of random processes on graphs that include the discrete Bak-Sneppen (DBS) process and the several versions of the contact process (CP), with a focus on the former. These processes are parametrized by a probability $0\le p\le 1$ that controls a local update rule. Numerical simulations reveal a phase transition when $p$ goes from 0 to 1. Analytically little is known about the phase transition threshold, even for one-dimensional chains. In this article we consider a power-series approach based on representing certain quantities, such as the survival probability or the expected number of steps per site to reach the steady state, as a power-series in $p$. We prove that the coefficients of those power series stabilize as the length $n$ of the chain grows. This is a phenomenon that has been used in the physics community but was not yet proven. We show that for local events $A,B$ of which the support is a distance $d$ apart we have $\mathrm{cor}(A,B)=\mathcal{O}(p^d)$. The stabilization allows for the (exact) computation of coefficients for arbitrary large systems which can then be analyzed using the wide range of existing methods of power series analysis.

25 pages, 5 figures