First passage times for threshold growth dynamics on \(\mathbb{Z}^ 2\). (English) Zbl 0872.60077

Summary: In the threshold growth model on an integer lattice, the occupied set grows according to a simple local rule: a site becomes occupied iff it sees at least a threshold number of already occupied sites in its prescribed neighborhood. We analyze the behavior of two-dimensional threshold growth dynamics started from a sparse Bernoulli density of occupied sites. We explain how nucleation of rare centers, invariant shapes and interaction between growing droplets influence the first passage time in the supercritical case. We also briefly address scaling laws for the critical case.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
52A10 Convex sets in \(2\) dimensions (including convex curves)
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[1] Adler, J. (1991). Bootstrap percolation. Phy s. A 171 453-470.
[2] Aizenman, M. and Lebowitz, J. (1988). Metastability effects in bootstrap percolation. J. Phy s. A: Math. Gen. 21 3801-3813. · Zbl 0656.60106
[3] Andjel, E. D. (1993). Characteristic exponents for two-dimensional bootstrap percolation. Ann. Prob. 21 926-935. · Zbl 0787.60120
[4] Andjel, E. D., Mountford, T. S. and Schonmann, R. H. (1995). Equivalence of exponential decay rates for bootstrap percolation like cellular automata. Ann. Inst. H. Poincaré 31 13-25. · Zbl 0813.60097
[5] Arratia, R., Goldstein, L. and Gordon, L. (1989). Two moments suffice for Poisson approximation: the Chen-Stein method. Ann. Probab. 17 9-25. · Zbl 0675.60017
[6] Barbour, A. D., Holst, L. and Janson, S. (1993). Poisson Approximation. Oxford Univ. Press. · Zbl 0784.60053
[7] Bohman, T. (1996). Unpublished manuscript.
[8] Durrett, R. (1988). Lecture Notes on Particle Sy stems and Percolation. Wadsworth and Brooks/Cole, Belmont, CA. · Zbl 0659.60129
[9] Durrett, R. and Griffeath, D. (1993). Asy mptotic behavior of excitable cellular automata. Experimental Math. 2 184-208. · Zbl 0796.60100
[10] Fisch, R., Gravner, J. and Griffeath, D. (1991). Threshold-range scaling for the excitable cellular automata. Statistics and Computing 1 23-39. · Zbl 0737.60088
[11] Gravner, J. and Griffeath, D. (1993). Threshold growth dy namics. Trans. Amer. Math. Soc. 340 837-870. JSTOR: · Zbl 0791.58053
[12] Kesten, H and Schonmann, R. H. (1995). On some growth models with a small parameter. Probab. Theory Related Fields 101 435-468. · Zbl 0820.60084
[13] Mountford, T. S. (1992). Rates for the probability of large cubes being noninternally spanned in modified bootstrap percolation. Probab. Theory Related Fields 93 174-193. · Zbl 0767.60102
[14] Mountford, T. S. (1995). Critical lengths for semi-oriented bootstrap percolation. Stochastic Process. Appl. 95 185-205. · Zbl 0821.60092
[15] Schonmann, R. H. (1990). Finite size scaling behavior of a biased majority rule cellular automaton. Phy s. A 167 619-627.
[16] Schonmann, R. H. (1990). Critical points of 2-dimensional bootstrap percolation-like cellular automata. J. Statist. Phy s. 58 1239-1244. · Zbl 0712.68071
[17] Schonmann, R. H. (1992). On the behavior of some cellular automata related to bootstrap percolation. Ann. Probab. 20 174-193. · Zbl 0742.60109
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