The Annals of Applied Probability

Unoriented first-passage percolation on the n-cube

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

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1476884298

Digital Object Identifier
doi:10.1214/15-AAP1155

Mathematical Reviews number (MathSciNet)
MR3563188

Zentralblatt MATH identifier
1353.60088

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

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

Citation

Martinsson, Anders. Unoriented first-passage percolation on the n -cube. Ann. Appl. Probab. 26 (2016), no. 5, 2597--2625. doi:10.1214/15-AAP1155. https://projecteuclid.org/euclid.aoap/1476884298


Export citation

References

  • [1] Aldous, D. (1989). Probability Approximations Via the Poisson Clumping Heuristic. Applied Mathematical Sciences 77. Springer, New York.
  • [2] 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.
  • [3] Durrett, R. (1988). Lecture Notes on Particle Systems and Percolation. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA.
  • [4] 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.
  • [5] Hegarty, P. and Martinsson, A. (2014). On the existence of accessible paths in various models of fitness landscapes. Ann. Appl. Probab. 24 1375–1395.
  • [6] Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.