Finite-Size Effects for Anisotropic Bootstrap Percolation: Logarithmic Corrections

Abstract

In this note we analyse an anisotropic, two-dimensional bootstrap percolation model introduced by Gravner and Griffeath. We present upper and lower bounds on the finite-size effects. We discuss the similarities with the semi-oriented model introduced by Duarte.

Metastability Thresholds for Anisotropic Bootstrap Percolation in Three Dimensions

“…Naively, one could suppose that we need a critical droplet with one face of supercritical size for the (1, 1) model, and two of supercritical size for the (1, c) model. However, we can do better than that, following essentially the same line of thought as in [17] which is similar to that in [2] and which was based on an unpublished observation of Roberto Schonmann. There a suitable critical droplet for the (1, 2) model was found to be a strip of length C p ln 1 p , and width 2, with C a large enough constant.…”

“…For the bound in the other direction, again a generalization of the method of [17] for the (1, 2) model is needed. First, we generalize (2) of [17] to (5.2) (this paper, Section 5.1). Then we repeat the calculation that leads to (5) of [17], replacing p 2 byp b and p byp a , and choosing x = C 2 1 p a ln 1 p and y = p −b+1/2 instead of choosing k = 1 p 3/2 and l = C 2 1 p ln 1 p .…”

“…First, we generalize (2) of [17] to (5.2) (this paper, Section 5.1). Then we repeat the calculation that leads to (5) of [17], replacing p 2 byp b and p byp a , and choosing x = C 2 1 p a ln 1 p and y = p −b+1/2 instead of choosing k = 1 p 3/2 and l = C 2 1 p ln 1 p . This will lead to the other inequality.…”

“…The following lemma bears resemblance to the derivation of (3.30) of [8], and of (23) of [23], but we derive the bound for the (a, b) model with arbitrary a and b. In the case (a, b) = (1, 2), one may choose x c < 1 p ln 1 p and y c < p −3/2 as in [17], but note that this choice for y c is not essential: the lemma is valid for any y c < p −2+ . For arbitrary r, we have…”

Higher order corrections for anisotropic bootstrap percolation

“…The anisotropic model was first studied by Gravner and Griffeath [28] in 1996. In 2007, the second and third authors [26] determined the correct order of magnitude of p c . More recently, the first and second authors [23] proved that the anisotropic model exhibits a sharp threshold by determining the first term in (1.3).…”