# Uniformity of hitting times of the contact process

## TL;DR

This paper proves uniformity properties of hitting times in the supercritical contact process on lattices, showing tightness and specific growth behaviors of infection times across sites, with implications for understanding infection spread dynamics.

## Contribution

It introduces new uniformity results for hitting times in the supercritical contact process, including stochastic tightness and growth properties, based on tightness of essential hitting times.

## Key findings

- The family of scaled differences of hitting times is stochastically tight.
- Existence of sites where infection times grow linearly with high probability.
- Hitting times exhibit uniformity and predictable growth patterns.

## Abstract

For the supercritical contact process on the hyper-cubic lattice started from a single infection at the origin and conditioned on survival, we establish two uniformity results for the hitting times $t(x)$, defined for each site $x$ as the first time at which it becomes infected. First, the family of random variables $(t(x)-t(y))/|x-y|$, indexed by $x \neq y$ in $\mathbb{Z}^d$, is stochastically tight. Second, for each $\varepsilon >0$ there exists $x$ such that, for infinitely many integers $n$, $t(nx) < t((n+1)x)$ with probability larger than $1-\varepsilon$. A key ingredient in our proofs is a tightness result concerning the essential hitting times of the supercritical contact process introduced by Garet and Marchand (Ann.\ Appl.\ Probab., 2012).