The Time of Bootstrap Percolation in High Dimensions

2025/05/16 by Zhu, Fengxing
Mathematics · Physics and Astronomy · #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2505.11410

openalex publication_date 2025/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the d-neighbor bootstrap percolation process on the d-dimensional torus, with vertex set V=\1,⋯,n\^d and edge set \xy:∑_i=1^d|x_i-y_i (mod n)|=1\. We determine the percolation time up to a constant factor with high probability when the initial infection probability is in a certain range and the infection threshold is d, extending one of the two main theorems from Balister,Bollobás, and Smith (2016) about the percolation time with the infection threshold equal to 2 on the two-dimensional torus.

Citations

• Bootstrap percolation on the high-dimensional Hamming graph
• Maximum percolation time in two-dimensional bootstrap percolation
• The time of bootstrap percolation in two dimensions
• The time of bootstrap percolation with dense initial sets for all thresholds
• The sharp threshold for bootstrap percolation in all dimensions
• Bootstrap percolation in high dimensions
• Majority bootstrap percolation on the hypercube
• The threshold regime of finite volume bootstrap percolation
• Sharp Metastability Threshold for Two-Dimensional Bootstrap Percolation
• Finite size scaling in three-dimensional bootstrap percolation
• Bootstrap percolation on a Bethe lattice

Related

• The time of bootstrap percolation with dense initial sets for all thresholds
• The time of bootstrap percolation for dense initial sets
• Maximum percolation time in two-dimensional bootstrap percolation
• The time of bootstrap percolation in two dimensions
• Explosive appearance of cores and bootstrap percolation on lattices