The Annals of Applied Probability

The size of the boundary in first-passage percolation

Michael Damron, Jack Hanson, and Wai-Kit Lam

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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 $t$.

Article information

Source
Ann. Appl. Probab., Volume 28, Number 5 (2018), 3184-3214.

Dates
Received: September 2017
Revised: January 2018
First available in Project Euclid: 28 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1535443246

Digital Object Identifier
doi:10.1214/18-AAP1388

Mathematical Reviews number (MathSciNet)
MR3847985

Zentralblatt MATH identifier
06974777

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35]

Keywords
First-passage percolation boundary Eden model limit shape

Citation

Damron, Michael; Hanson, Jack; Lam, Wai-Kit. The size of the boundary in first-passage percolation. Ann. Appl. Probab. 28 (2018), no. 5, 3184--3214. doi:10.1214/18-AAP1388. https://projecteuclid.org/euclid.aoap/1535443246


Export citation

References

  • [1] Alexander, K. S. (1997). Approximation of subadditive functions and convergence rates in limiting-shape results. Ann. Probab. 25 30–55.
    Zentralblatt MATH: 0882.60090
    Digital Object Identifier: doi:10.1214/aop/1024404277
    Project Euclid: euclid.aop/1024404277
  • [2] Antal, P. and Pisztora, A. (1996). On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24 1036–1048.
    Zentralblatt MATH: 0871.60089
    Digital Object Identifier: doi:10.1214/aop/1039639377
    Project Euclid: euclid.aop/1039639377
  • [3] Auffinger, A., Damron, M. and Hanson, J. (2017). 50 Years of First-Passage Percolation. University Lecture Series 68. Amer. Math. Soc., Providence, RI.
    Zentralblatt MATH: 06828688
  • [4] Bouch, G. (2015). The expected perimeter in Eden and related growth processes. J. Math. Phys. 56 Article ID 123302.
    Zentralblatt MATH: 1337.82016
    Digital Object Identifier: doi:10.1063/1.4936210
  • [5] Boucheron, S., Lugosi, G. and Massart, P. (2013). Concentration Inequalities. A Nonasymptotic Theory of Independence. Oxford Univ. Press, Oxford.
    Zentralblatt MATH: 1279.60005
  • [6] Burdzy, K. and Pal, S. (2016). Twin peaks. Available at arXiv:1606.08025.
    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.
    Zentralblatt MATH: 0462.60012
    Digital Object Identifier: doi:10.1214/aop/1176994364
    Project Euclid: euclid.aop/1176994364
  • [8] Damron, M., Lam, W.-K. and Wang, X. (2017). Asymptotics for $2D$ critical first passage percolation. Ann. Probab. 45 2941–2970.
    Zentralblatt MATH: 1378.60115
    Digital Object Identifier: doi:10.1214/16-AOP1129
    Project Euclid: euclid.aop/1506132031
  • [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.
  • [10] Grimmett, G. (1999). Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 321. Springer, Berlin.
    Zentralblatt MATH: 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. A 18 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.
    Zentralblatt MATH: 0848.60089
  • [14] Richardson, D. (1973). Random growth in a tessellation. Math. Proc. Cambridge Philos. Soc. 74 515–528.
    Zentralblatt MATH: 0295.62094
    Digital Object Identifier: doi:10.1017/S0305004100077288
  • [15] Timár, Á. (2013). Boundary-connectivity via graph theory. Proc. Amer. Math. Soc. 141 475–480.
  • [16] Zabolitzky, J. G. and Stauffer, D. (1986). Simulation of large Eden clusters. Phys. Rev. A 34 1523–1530.

The Institute of Mathematical Statistics

The Annals of Applied Probability

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more