# A discontinuous phase transition in the threshold-$\theta \geq 2$ contact process on random graphs

## TL;DR

This paper demonstrates a discontinuous phase transition in the threshold-$ heta$ contact process on random graphs, showing how infection persistence sharply changes with infection probability, depending on the graph's degree distribution.

## Contribution

It proves the existence of a discontinuous phase transition in the threshold-$ heta$ contact process on random graphs with general degree distributions, answering a question by Chatterjee and Durrett.

## Key findings

- High infection probability leads to long-term survival of the process.
- Low initial density results in rapid die-out.
- The process's behavior varies sharply at a critical infection probability.

## Abstract

We study the discrete-time threshold-$\theta \geq 2$ contact process on random graphs of general degrees. For random graphs with a given degree distribution $\mu$, we show that if $\mu$ is lower bounded by $\theta+2$ and has finite $k$th moments for all $k>0$, then the discrete-time threshold-$\theta$ contact process on the random graph exhibits a discontinuous phase transition in the emergence of metastability, thus answering a question of Chatterjee and Durrett \cite{cd13}. To be specific, we establish that (i) for any large enough infection probability $p>p_1$, the process started from the all-infected state w.h.p. survives for $e^{\Theta(n)}$-time, maintaining a large density of infection; (ii) for any $p<1$, if the initial density is smaller than $\varepsilon(p)>0$, then it dies out in $O(\log n)$-time w.h.p.. We also explain some extensions to more general random graphs, including the Erd\H{o}s-R\'enyi graphs. Moreover, we prove that the threshold-$\theta$ contact process on a random $(\theta+1)$-regular graph dies out in time $n^{O(1)}$ w.h.p..