The Annals of Probability

First-passage percolation on Cartesian power graphs

Anders Martinsson

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

We consider first-passage percolation on the class of “high-dimensional” graphs that can be written as an iterated Cartesian product $G\square G\square\dots\square G$ of some base graph $G$ as the number of factors tends to infinity. We propose a natural asymptotic lower bound on the first-passage time between $(v,v,\dots,v)$ and $(w,w,\dots,w)$ as $n$, the number of factors, tends to infinity, which we call the critical time $t^{*}_{G}(v,w)$. Our main result characterizes when this lower bound is sharp as $n\rightarrow\infty$. As a corollary, we are able to determine the limit of the so-called diagonal time-constant in $\mathbb{Z}^{n}$ as $n\rightarrow\infty$ for a large class of distributions of passage times.

Article information

Source
Ann. Probab., Volume 46, Number 2 (2018), 1004-1041.

Dates
Received: October 2015
Revised: April 2017
First available in Project Euclid: 9 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1520586274

Digital Object Identifier
doi:10.1214/17-AOP1199

Mathematical Reviews number (MathSciNet)
MR3773379

Zentralblatt MATH identifier
06864078

Subjects
Primary: 60C05: Combinatorial probability
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35]

Keywords
First-passage percolation power graph high dimension time constant hypercube

Citation

Martinsson, Anders. First-passage percolation on Cartesian power graphs. Ann. Probab. 46 (2018), no. 2, 1004--1041. doi:10.1214/17-AOP1199. https://projecteuclid.org/euclid.aop/1520586274


Export citation

References

  • [1] Aldous, D. (1989). Probability Approximations Via the Poisson Clumping Heuristic. Applied Mathematical Sciences 77. Springer, New York.
  • [2] Alon, N. and Spencer, J. H. (2008). The Probabilistic Method, 3rd ed. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, Hoboken, NJ.
  • [3] Auffinger, A. and Tang, S. (2016). On the time constant of high dimensional first passage percolation. Electron. J. Probab. 21 Paper No. 24, 23.
  • [4] Blair-Stahn, N. D. First passage percolation and competition models. Preprint. Available at arXiv:1005.0649.
  • [5] Bollobás, B. and Kohayakawa, Y. (1997). On Richardson’s model on the hypercube. In Combinatorics, Geometry and Probability (Cambridge, 1993) 129–137. Cambridge Univ. Press, Cambridge.
  • [6] Couronné, O., Enriquez, N. and Gerin, L. (2011). Construction of a short path in high-dimensional first passage percolation. Electron. Commun. Probab. 16 22–28.
  • [7] Cox, J. T. and Durrett, R. (1983). Oriented percolation in dimensions $d\geq 4$: Bounds and asymptotic formulas. Math. Proc. Cambridge Philos. Soc. 93 151–162.
  • [8] Dhar, D. (1988). First passage percolation in many dimensions. Phys. Lett. A 130 308–310.
  • [9] Dhar, D. (1986). Asymptotic shape of Eden clusters. In On Growth and Form. Fractal and Non-Fractal Patterns in Physics. (H. E. Stanley and N. Ostrowsky, eds.). NATO ASI Series 100 288–292. Springer, Berlin.
  • [10] Fill, J. A. and Pemantle, R. (1993). Percolation, first-passage percolation and covering times for Richardson’s model on the $n$-cube. Ann. Appl. Probab. 3 593–629.
  • [11] Grimmett, G. (1999). Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 321. Springer, Berlin.
  • [12] 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.
  • [13] Li, L. (2017). Phase transition for accessibility percolation on hypercubes. J. Theoret. Probab. Available at https://doi.org/10.1007/s10959-017-0769-x.
  • [14] Martinsson, A. Accessibility percolation and first-passage site percolation on the unoriented binary hypercube. Preprint. Available at arXiv:1501.02206.
  • [15] Martinsson, A. (2016). Unoriented first-passage percolation on the $n$-cube. Ann. Appl. Probab. 26 2597–2625.