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

## TL;DR

This paper develops a power-series method to analyze phase transitions in local graph processes like the Bak-Sneppen and contact processes, proving coefficient stabilization and decay of correlations with distance.

## Contribution

It introduces a novel power-series approach to study phase transitions in local processes on graphs and proves coefficient stabilization as system size grows.

## Key findings

- Coefficients of power series stabilize with increasing chain length
- Correlation between distant local events decays as p^d
- Method enables exact computation for large systems

## 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 $0\leq p \leq 1$ 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.