The Annals of Applied Probability

Unoriented first-passage percolation on the n-cube

Anders Martinsson

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The $n$-dimensional binary hypercube is the graph whose vertices are the binary $n$-tuples $\{0,1\}^{n}$ and where two vertices are connected by an edge if they differ at exactly one coordinate. We prove that if the edges are assigned independent mean 1 exponential costs, the minimum length $T_{n}$ of a path from $(0,0,\dots,0)$ to $(1,1,\dots,1)$ converges in probability to $\ln(1+\sqrt{2})\approx0.881$. It has previously been shown by Fill and Pemantle [Ann. Appl. Probab. 3 (1993) 593–629] that this so-called first-passage time asymptotically almost surely satisfies $\ln(1+\sqrt{2})-o(1)\leq T_{n}\leq1+o(1)$, and has been conjectured to converge in probability by Bollobás and Kohayakawa [In Combinatorics, Geometry and Probability (Cambridge, 1993) (1997) 129–137 Cambridge]. A key idea of our proof is to consider a lower bound on Richardson’s model, closely related to the branching process used in the article by Fill and Pemantle to obtain the bound $T_{n}\geq\ln(1+\sqrt{2})-o(1)$. We derive an explicit lower bound on the probability that a vertex is infected at a given time. This result is formulated for a general graph and may be applicable in a more general setting.

Article information

Ann. Appl. Probab., Volume 26, Number 5 (2016), 2597-2625.

Received: June 2014
Revised: January 2015
First available in Project Euclid: 19 October 2016

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Zentralblatt MATH identifier

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

First-passage percolation Richardson’s model hypercube branching translation process lower bound on Richardson’s model


Martinsson, Anders. Unoriented first-passage percolation on the n -cube. Ann. Appl. Probab. 26 (2016), no. 5, 2597--2625. doi:10.1214/15-AAP1155.

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