First contact percolation

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.

💡 Research Summary

The paper introduces a novel stochastic growth model on the integer lattice ℤᵈ, called First Contact Percolation (FCP), which replaces the usual scalar passage times of First Passage Percolation (FPP) with random closed subsets of ℝ representing “contact times” on each edge. An infection starts at the origin and can traverse an edge only at a time belonging to the edge’s contact set, and the sequence of times along a path must be non‑decreasing. Consequently the induced travel time D₀(x,y) is generally asymmetric and satisfies only a weak triangle inequality.

The authors first treat the case where the contact sets are stationary under real‑line shifts. They define a distribution μ on