# Bootstrap percolation in random $k$-uniform hypergraphs

## TL;DR

This paper studies bootstrap percolation on random hypergraphs, identifying thresholds for widespread infection depending on initial infected set size, with exponential decay of failure probability for the case of graphs.

## Contribution

It establishes a threshold phenomenon for infection spread in hypergraphs and provides exponential bounds for the graph case, advancing understanding of percolation dynamics.

## Key findings

- Threshold for infection spread depends on initial infected set size.
- Below threshold, only a few vertices become infected.
- Above threshold, almost all vertices become infected.

## Abstract

We investigate bootstrap percolation with infection threshold $r> 1$ on the binomial $k$-uniform random hypergraph $H_k(n,p)$ in the regime $n^{-1}\ll n^{k-2}p \ll n^{-1/r}$, when the initial set of infected vertices is chosen uniformly at random from all sets of given size. We establish a threshold such that if there are less vertices in the initial set of infected vertices, then whp only a few additional vertices become infected, while if the initial set of infected vertices exceeds the threshold then whp almost every vertex becomes infected. In addition, for $k=2$, we show that the probability of failure decreases exponentially.