Long-range contact process and percolation on a random lattice
Pablo A. Gomes, Bernardo N. B. de Lima
TL;DR
The paper addresses phase transitions in two interconnected models: CPDR, a long-range contact process with dynamically updated random infection ranges, and APRR, an anisotropic long-range oriented percolation model with random ranges. By coupling CPDR to subcritical branching processes and employing block renormalization to compare to oriented percolation, it establishes extinction criteria for small infection rates when E[N^d] is finite and survival when tail conditions on N hold. In the percolation setting, finite EN yields a positive critical curve q_c(p) for p < 1, while infinite EN with heavy-tailed tails drives q_c(p) to zero for all p > 0; in one dimension, sharp beta thresholds govern percolation when q = 0. Overall, the results reveal that the tail behavior of the random infection range N decisively controls persistence versus extinction or percolation, highlighting how random dynamical environments shape long-range interactions.
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 $N$.