Long-range contact process and percolation on a random lattice

TL;DR

This paper investigates phase transition phenomena in long-range oriented percolation and contact processes on random lattices, analyzing how the distribution of interaction ranges influences critical behavior.

Contribution

It introduces models with dynamically assigned and i.i.d. random ranges, establishing phase transition results based on the distribution $N$ for both models.

Findings

Existence of phase transition depending on the distribution $N$

Results applicable to models with independent, dynamically updated ranges

Extension of percolation theory to long-range, random-lattice settings

Abstract

We study the phase transition phenomena for long-range oriented percolation and contact process. We studied a contact process in which the range of each vertex are independent, updated dynamically and given by some distribution $N$. We also study an analogous oriented percolation model on the hyper-cubic lattice, here there is a special direction where long-range oriented bonds are allowed; the range of all vertices are given by an i.i.d. sequence of random variables with common distribution $N$. For both models, we prove some results about the existence of a phase transition in terms of the distribution $N$.