# Smallest percolating sets in bootstrap percolation on grids

## TL;DR

This paper establishes the minimal size of initial infected sets needed for percolation in d-dimensional grid bootstrap models, and bounds the percolation time, filling a key gap in extremal bootstrap percolation theory.

## Contribution

It proves that the smallest percolating sets in d-dimensional grids have size n^{d-1} and percolate within a time proportional to n^2, for all dimensions d.

## Key findings

- Smallest percolating sets have size n^{d-1}
- Percolation occurs within time at most c_d n^2
- Fills a fundamental gap in bootstrap percolation literature

## Abstract

In this paper we fill in a fundamental gap in the extremal bootstrap percolation literature, by providing the first proof of the fact that for all $d \geq 1$, the size of the smallest percolating sets in $d$-neighbour bootstrap percolation on $[n]^d$, the $d$-dimensional grid of size $n$, is $n^{d-1}$. Additionally, we prove that such sets percolate in time at most $c_d n^2$, for some constant $c_d >0 $ depending on $d$ only.