Local survival of spread of infection among biased random walks

TL;DR

This paper investigates how infection spreads among biased random walks on a lattice, showing that infection can either die out locally or spread globally depending on the initial particle density, using genealogical and renormalization techniques.

Contribution

It provides a detailed analysis of infection spread dynamics among biased random walks, introducing new methods for small and large density regimes.

Findings

Infection does not survive locally at low densities.

At high densities, infection spreads throughout the entire lattice.

Different analytical techniques are used for different density regimes.

Abstract

We study infection spread among biased random walks on $\mathbb{Z}^{d}$. The random walks move independently and an infected particle is placed at the origin at time zero. Infection spreads instantaneously when particles share the same site and there is no recovery. If the initial density of particles is small enough, the infected cloud travels in the direction of the bias of the random walks, implying that the infection does not survive locally. When the density is large, the infection spreads to the whole $\mathbb{Z}^{d}$. The proofs rely on two different techniques. For the small density case, we use a description of the infected cloud through genealogical paths, while the large density case relies on a renormalization scheme.