The Annals of Probability

First-passage percolation on Cartesian power graphs

Anders Martinsson

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

First-passage percolation power graph high dimension time constant hypercube


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

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  • [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
  • [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.