The Interaction Light Cone of the Discrete Bak–Sneppen, Contact and other local processes

We consider a class of random processes on graphs that include the discrete Bak–Sneppen process and several versions of the contact process, with a focus on the former. These processes are parametrized by a probability \(0\le p \le 1\) that controls a local update rule, and often exhibit a phase transition in this parameter. In general, analyzing properties of the phase transition is challenging, 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 for various families of graphs, including the family of chain graphs. This phenomenon has been used in the physics community but was not yet proven. We also show that for local events A, B of which the support is a distance d apart we have \({\mathrm {cor}}(A,B) = O(p^d)\). 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.