Deterministic bootstrap percolation in high dimensional grids

2013/08/30 by Hao Huang, Huang, Hao, Choongbum Lee +1
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics #math.CO

paper · pdf · doi:10.48550/arxiv.1308.6791

This paper has been withdrawn by the authors since Theorem 3.1 has been proved in an earlier paper

openalex publication_date 2013/08/30 · arxiv created 2013/09/04 · arxiv updated 2013/09/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Bootstrap percolation on the random regular graph

Related

• Smallest percolating sets in bootstrap percolation on grids
• Bootstrap percolation in high dimensions
• Maximum percolation time in two-dimensional bootstrap percolation
• The Time of Bootstrap Percolation in High Dimensions
• The sharp threshold for bootstrap percolation in all dimensions