The $d$-dimensional bootstrap percolation models with threshold at least double exponential

Abstract

Consider a p-random subset A of initially infected vertices in the discrete cube [L] d , and assume that the neighbourhood of each vertex consists of the a i nearest neighbours in the ±e i -directions for each i ∈ {1, 2, . . . , d}, where a 1 a 2 . . . a d . Suppose we infect any healthy vertex v ∈ [L] d already having r infected neighbours, and that infected sites remain infected forever. In this paper we determine the (d − 1)-times iterated logarithm of the critical length for percolation up to a constant factor, for all d-tuples (a 1 , . . . , a d ) and all r ∈ {a 2Moreover, we reduce the problem of determining this (coarse) threshold for all d 3 and all r ∈ {a d + 1, . . . , a 1 + a 2 + • • • + a d }, to that of determining the threshold for all d 3 and all r ∈ {a d + 1, . . . , a d−1 + a d }. L c (N a 1 ,...,a d r , p) := min{L ∈ N : P p ([A] = [L] d ) 1/2}.

Three-dimensional 2-critical bootstrap percolation: The stable sets approach

“…The threshold was also determined in the general case r = a + b by van Enter and Fey [14] and the proof can be extended to all b + 1 r a + b: These models were studied by van Enter and Fey [14], and the present author [4] for r ∈ {1 + b + c, . .…”