Open Access
October 2016 Unoriented first-passage percolation on the n-cube
Anders Martinsson
Ann. Appl. Probab. 26(5): 2597-2625 (October 2016). DOI: 10.1214/15-AAP1155

Abstract

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.

Citation

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Anders Martinsson. "Unoriented first-passage percolation on the n-cube." Ann. Appl. Probab. 26 (5) 2597 - 2625, October 2016. https://doi.org/10.1214/15-AAP1155

Information

Received: 1 June 2014; Revised: 1 January 2015; Published: October 2016
First available in Project Euclid: 19 October 2016

zbMATH: 1353.60088
MathSciNet: MR3563188
Digital Object Identifier: 10.1214/15-AAP1155

Subjects:
Primary: 60K35
Secondary: 60C05

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

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.26 • No. 5 • October 2016
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