Deterministic bootstrap percolation in high dimensional grids

TL;DR

This paper analyzes the minimum initial vertices needed for percolation in high-dimensional grids under k-neighbor bootstrap percolation, confirming a conjecture and providing precise asymptotic bounds.

Contribution

It establishes the exact asymptotic behavior of the percolation threshold in high-dimensional grids, confirming a conjecture of Pete.

Findings

Minimum initial vertices for percolation is (1-d/k)n^d + O(n^{d-1})

Results confirm a conjecture of Pete

Provides precise asymptotic bounds for high-dimensional bootstrap percolation

Abstract

In this paper, we study the k-neighbor bootstrap percolation process on the d-dimensional grid [n]^d, and show that the minimum number of initial vertices that percolate is (1-d/k)n^d + O(n^{d-1})$ when d<=k<=2d. This confirms a conjecture of Pete.