# zbMATH — the first resource for mathematics

The size of the boundary in first-passage percolation. (English) Zbl 1402.60127
Summary: First-passage percolation is a random growth model defined using i.i.d. edge-weights $$(t_{e})$$ on the nearest-neighbor edges of $$\mathbb{Z}^{d}$$. An initial infection occupies the origin and spreads along the edges, taking time $$t_{e}$$ to cross the edge $$e$$. In this paper, we study the size of the boundary of the infected (“wet”) region at time $$t$$, $$B(t)$$. It is known that $$B(t)$$ grows linearly, so its boundary $$\partial B(t)$$ has size between $$ct^{d-1}$$ and $$Ct^{d}$$. Under a weak moment condition on the weights, we show that for most times, $$\partial B(t)$$ has size of order $$t^{d-1}$$ (smooth). On the other hand, for heavy-tailed distributions, $$B(t)$$ contains many small holes, and consequently we show that $$\partial B(t)$$ has size of order $$t^{d-1+\alpha }$$ for some $$\alpha >0$$ depending on the distribution. In all cases, we show that the exterior boundary of $$B(t)$$ [edges touching the unbounded component of the complement of $$B(t)$$] is smooth for most times. Under the unproven assumption of uniformly positive curvature on the limit shape for $$B(t)$$, we show the inequality $$\#\partial B(t)\leq (\log t)^{C}t^{d-1}$$ for all large $$t$$.
##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B43 Percolation
##### Keywords:
first-passage percolation; boundary; Eden model; limit shape
Full Text:
##### References:
 [1] Alexander, K. S. (1997). Approximation of subadditive functions and convergence rates in limiting-shape results. Ann. Probab.25 30–55. · Zbl 0882.60090 [2] Antal, P. and Pisztora, A. (1996). On the chemical distance for supercritical Bernoulli percolation. Ann. Probab.24 1036–1048. · Zbl 0871.60089 [3] Auffinger, A., Damron, M. and Hanson, J. (2017). 50 Years of First-Passage Percolation. University Lecture Series68. Amer. Math. Soc., Providence, RI. · Zbl 1452.60002 [4] Bouch, G. (2015). The expected perimeter in Eden and related growth processes. J. Math. Phys.56 Article ID 123302. · Zbl 1337.82016 [5] Boucheron, S., Lugosi, G. and Massart, P. (2013). Concentration Inequalities. A Nonasymptotic Theory of Independence. Oxford Univ. Press, Oxford. · Zbl 1279.60005 [6] Burdzy, K. and Pal, S. (2016). Twin peaks. Available at arXiv:1606.08025. [7] Cox, J. T. and Durrett, R. (1981). Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Probab.9 583–603. · Zbl 0462.60012 [8] Damron, M., Lam, W.-K. and Wang, X. (2017). Asymptotics for $$2D$$ critical first passage percolation. Ann. Probab.45 2941–2970. · Zbl 1378.60115 [9] Eden, M. (1961). A two-dimensional growth process. In Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. IV 223–239. Univ. California Press, Berkeley, CA. · Zbl 0104.13801 [10] Grimmett, G. (1999). Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 321. Springer, Berlin. · Zbl 0926.60004 [11] Kesten, H. (1986). Aspects of first passage percolation. In École d’Été de Probabilités de Saint-Flour XIV—1984. Lecture Notes in Math.1180 125–264. Springer, Berlin. [12] Leyvraz, F. (1985). The “active perimeter” in cluster growth models: A rigorous bound. J. Phys. A18 L941–L945. [13] Newman, C. M. (1995). A surface view of first-passage percolation. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) 1017–1023. Birkhäuser, Basel. · Zbl 0848.60089 [14] Richardson, D. (1973). Random growth in a tessellation. Math. Proc. Cambridge Philos. Soc.74 515–528. · Zbl 0295.62094 [15] Timár, Á. (2013). Boundary-connectivity via graph theory. Proc. Amer. Math. Soc.141 475–480. · Zbl 1259.05049 [16] Zabolitzky, J.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.