# Multi-scale Lipschitz percolation of increasing events for Poisson random walks

## TL;DR

This paper develops a multi-scale Lipschitz percolation framework for increasing events in Poisson random walks on weighted lattices, enabling proofs of phenomena like infection spread with positive speed.

## Contribution

It introduces a novel Lipschitz percolation method for Poisson random walks, facilitating multi-scale analysis of increasing events in this stochastic setting.

## Key findings

- Existence of Lipschitz surfaces separating the origin from infinity.
- Application to proving positive speed of infection spread.
- Robust framework for multi-scale probabilistic analysis.

## Abstract

Consider the graph induced by $\mathbb{Z}^d$, equipped with uniformly elliptic random conductances. At time $0$, place a Poisson point process of particles on $\mathbb{Z}^d$ and let them perform independent simple random walks. Tessellate the graph into cubes indexed by $i\in\mathbb{Z}^d$ and tessellate time into intervals indexed by $\tau$. Given a local event $E(i,\tau)$ that depends only on the particles inside the space time region given by the cube $i$ and the time interval $\tau$, we prove the existence of a Lipschitz connected surface of cells $(i,\tau)$ that separates the origin from infinity on which $E(i,\tau)$ holds. This gives a directly applicable and robust framework for proving results in this setting that need a multi-scale argument. For example, this allows us to prove that an infection spreads with positive speed among the particles.