Sharp metastability threshold for two-dimensional bootstrap percolation

By Alexander E. Holroyd. 2002

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Abstract. In the bootstrap percolation model, sites in an L by L square are initially independently declared active with probability p. At each time step, an inactive site becomes active if at least two of its four neighbours are active. We study the behaviour as p→0 and L→∞ simultaneously of the probability I(L,p) that the entire square is eventually active. We prove that I(L,p)→1 if , and I(L,p)→0 if , where λ=π2/18. We prove the same behaviour, with the same threshold λ, for the probability J(L,p) that a site is active by time L in the process on the infinite lattice. The same results hold for the so-called modified bootstrap percolation model, but with threshold λ′=π2/6. The existence of the thresholds λ,λ′ settles a conjecture of Aizenman and Lebowitz [3], while the determination of their values corrects numerical predictions of Adler, Stauffer and Aharony [2].

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ffa486502a03bf03cc5cfff605456f35bd27839e

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Citations in the corpus

RESILIENCE TO CONTAGION IN FINANCIAL NETWORKS (2011, unknown): “The bootstrap percolation process1 is a diffusion model that has been studied on a variety of graphs (see, e.g., Holroyd 2003; Balogh and Bollobás 2006; Balogh and Pittel 2007; Amini 2010a, b).”
The sharp threshold for bootstrap percolation in all dimensions (2010, methodology): “Motivated by these physical models, we shall consider bootstrap percolation on the grid [n]d = {1, . . . , n}d, and an initial set A ⊂ V (G) whose elements are chosen independently at random, each with probability p.”
Integrals, partitions, and cellular automata (2003, methodology): “The argument is a modification of that in [12].”
Recent advances in percolation theory and its applications (2015, background): “determined by Holroyd [221], for k = 2 on a finite 2D lattice Z(2) of linear size L, who proved”
Probability on Trees and Networks (2017, background): “(This is Proposition 13 of Hoffman, Holroyd, and Peres (2006).) Find a flow θ ≤ q.”
Bootstrap percolation on the hypercube (2006, methodology): “(This simple but useful idea was used in earlier versions of this paper and in a write-up of our results in [5]; later it was also discovered independently by Holroyd [20].”
Bootstrap percolation in three dimensions (2008, background): “When the set A is chosen at random, the main aim is to determine the critical probability pc(G,r) at which percolation (infection of the entire graph) becomes likely to occur.”
Kinetically Constrained Models (2010, unknown): “However, the results in (Aizenman and Lebowitz, 1988; Holroyd, 2003) also give the typical size of the incipient blocked cluster to which any fixed site belongs and that has to be eroded via successive moves from its boundary before the site in qustion can be unblocked.”