On the number and size of holes in the growing ball of first-passage percolation
HTML articles powered by AMS MathViewer
- by Michael Damron, Julian Gold, Wai-Kit Lam and Xiao Shen;
- Trans. Amer. Math. Soc. 377 (2024), 1641-1670
- DOI: https://doi.org/10.1090/tran/9035
- Published electronically: December 22, 2023
- HTML | PDF | Request permission
Abstract:
First-passage percolation is a random growth model defined on $\mathbb {Z}^d$ using i.i.d. nonnegative weights $(\tau _e)$ on the edges. Letting $T(x,y)$ be the distance between vertices $x$ and $y$ induced by the weights, we study the random ball of radius $t$ centered at the origin, $\mathbf {B}(t) = \{x \in \mathbb {Z}^d : T(0,x) \leq t\}$. It is known that for all such $\tau _e$, the number of vertices (volume) of $\mathbf {B}(t)$ is at least order $t^d$, and under mild conditions on $\tau _e$, this volume grows like a deterministic constant times $t^d$. Defining a hole in $\mathbf {B}(t)$ to be a bounded component of the complement $\mathbf {B}(t)^c$, we prove that if $\tau _e$ is not deterministic, then a.s., for all large $t$, $\mathbf {B}(t)$ has at least $ct^{d-1}$ many holes, and the maximal volume of any hole is at least $c\log t$. Conditionally on the (unproved) uniform curvature assumption, we prove that a.s., for all large $t$, the number of holes is at most $(\log t)^C t^{d-1}$, and for $d=2$, no hole in $\mathbf {B}(t)$ has volume larger than $(\log t)^C$. Without curvature, we show that no hole has volume larger than $Ct \log t$.References
- Antonio Auffinger, Michael Damron, and Jack Hanson, 50 years of first-passage percolation, University Lecture Series, vol. 68, American Mathematical Society, Providence, RI, 2017. MR 3729447, DOI 10.1090/ulect/068
- Ronnie Barequet, Gill Barequet, and Günter Rote, Formulae and growth rates of high-dimensional polycubes, Combinatorica 30 (2010), no. 3, 257–275. MR 2728490, DOI 10.1007/s00493-010-2448-8
- G. Bouch, The expected perimeter in Eden and related growth processes, J. Math. Phys. 56.
- Raphaël Cerf and Marie Théret, Weak shape theorem in first passage percolation with infinite passage times, Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016), no. 3, 1351–1381 (English, with English and French summaries). MR 3531712, DOI 10.1214/15-AIHP686
- Michael Damron, Jack Hanson, and Wai-Kit Lam, The size of the boundary in first-passage percolation, Ann. Appl. Probab. 28 (2018), no. 5, 3184–3214. MR 3847985, DOI 10.1214/18-AAP1388
- Michael Damron, Jack Hanson, and Philippe Sosoe, Subdiffusive concentration in first-passage percolation, Electron. J. Probab. 19 (2014), no. 109, 27. MR 3286463, DOI 10.1214/EJP.v19-3680
- Michael Damron and Naoki Kubota, Rate of convergence in first-passage percolation under low moments, Stochastic Process. Appl. 126 (2016), no. 10, 3065–3076. MR 3542626, DOI 10.1016/j.spa.2016.04.001
- Michael Damron, Wai-Kit Lam, and Xuan Wang, Asymptotics for $2D$ critical first passage percolation, Ann. Probab. 45 (2017), no. 5, 2941–2970. MR 3706736, DOI 10.1214/16-AOP1129
- Michael Damron, Firas Rassoul-Agha, and Timo Seppäläinen, Random growth models, Notices Amer. Math. Soc. 63 (2016), no. 9, 1004–1008. MR 3525713, DOI 10.1090/noti1400
- Harry Kesten, Aspects of first passage percolation, École d’été de probabilités de Saint-Flour, XIV—1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, 1986, pp. 125–264. MR 876084, DOI 10.1007/BFb0074919
- F. Leyvraz, The “active perimeter” in cluster growth models: a rigorous bound, J. Phys. A. 18, L941–L945.
- Fedor Manin, Érika Roldán, and Benjamin Schweinhart, Topology and local geometry of the Eden model, Discrete Comput. Geom. 69 (2023), no. 3, 771–799. MR 4555869, DOI 10.1007/s00454-022-00474-w
- C. M. Newman, A surface view of first-passage percolation, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, pp. 1017–1023.
- Ádám Timár, Boundary-connectivity via graph theory, Proc. Amer. Math. Soc. 141 (2013), no. 2, 475–480. MR 2996951, DOI 10.1090/S0002-9939-2012-11333-4
Bibliographic Information
- Michael Damron
- Affiliation: School of Mathematics, Georgia Institute of Technology, 686 Cherry St., Atlanta, Georgia 30332
- MR Author ID: 883682
- Email: mdamron6@protonmail.com
- Julian Gold
- Affiliation: Center for Statistics and Machine Learning, Princeton University, 26 Prospect Ave., Princeton, New Jersey 08544
- MR Author ID: 1073993
- Email: julian.thomas.gold@gmail.com
- Wai-Kit Lam
- Affiliation: Department of Mathematics, National Taiwan University, Astronomy Mathematics Building 5F, No. 1, Sec. 4, Roosevelt Rd., Taipei 10617, Taiwan (R.O.C.)
- MR Author ID: 1236212
- ORCID: 0000-0001-6282-3561
- Email: waikitlam@ntu.edu.tw
- Xiao Shen
- Affiliation: Department of Mathematics, University of Utah, 155 South 1400 East, JWB 233, Salt Lake City, Utah 84112
- Email: xiao.shen@utah.edu
- Received by editor(s): May 20, 2022
- Received by editor(s) in revised form: January 24, 2023
- Published electronically: December 22, 2023
- Additional Notes: The research of the first author was supported by an NSF grant DMS-2054559 and an NSF CAREER award.
The research of the second author was supported by NSF Postdoctoral Research Fellowship DMS-1803622 while at Northwestern. The second author gratefully acknowledges financial support from the Schmidt DataX Fund at Princeton University made possible through a major gift from the Schmidt Futures Foundation.
The research of the third author was supported by the National Science and Technology Council in Taiwan Grant 110-2115-M002-012-MY3 and NTU New Faculty Founding Research Grant NTU-111L7452. - © Copyright 2023 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 377 (2024), 1641-1670
- MSC (2020): Primary 60K35
- DOI: https://doi.org/10.1090/tran/9035
- MathSciNet review: 4744738