First contact percolation

TL;DR

This paper introduces a novel contact percolation model based on contact times represented by random sets, establishing shape theorems and analyzing infection spread under stationary and periodic contact patterns.

Contribution

It develops a new contact percolation framework using contact sets, proving shape theorems and comparing infection speeds across different models.

Findings

Shape theorems for stationary contact times.

Universal limiting shapes for periodic contact times.

Less randomness increases infection speed.

Abstract

We study a version of first passage percolation on $\mathbb{Z}^d$ where the random passage times on the edges are replaced by contact times represented by random closed sets on $\mathbb{R}$. Similarly to the contact process without recovery, an infection can spread into the system along increasing sequences of contact times. In case of stationary contact times, we can identify associated first passage percolation models, which in turn establish shape theorems also for first contact percolation. In case of periodic contact times that reflect some reoccurring daily pattern, we also present shape theorems with limiting shapes that are universal with respect to the within-one-day contact distribution. In this case, we also prove a Poisson approximation for increasing numbers of within-one-day contacts. Finally, we present a comparison of the limiting speeds of three models -- all calibrated to have one expected contact per day -- that suggests that less randomness is beneficial for the speed of the infection. The proofs rest on coupling and subergodicity arguments.