Contact process on one-dimensional long-range percolation

TL;DR

This paper demonstrates that one-dimensional long-range percolation with a high exponent satisfies conditions for a positive critical value in the contact process, indicating a non-trivial phase transition.

Contribution

It shows that certain one-dimensional long-range percolation models meet criteria for phase transition in the contact process, extending previous theoretical conditions.

Findings

High exponent long-range percolation satisfies the cumulatively merged partition condition.

The contact process on this model has a positive critical value.

The model exhibits a non-trivial phase transition.

Abstract

Recently, by introducing the notion of cumulatively merged partition, M\'enard and Singh provide a sufficient condition on graphs ensuring that the critical value of the contact process is positive. In this note, we show that the one-dimensional long range percolation with high exponent satisfies their condition and thus the contact process exhibits a non-trivial phase transition.