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Site Percolation on the d-Dimensional Hamming Torus

Published online by Cambridge University Press:  29 November 2013

DAVID SIVAKOFF
DAVID SIVAKOFF*
Affiliation:
Department of Statistics, Department of Mathematics, Ohio State University, Columbus, OH 43210, USA (e-mail: [email protected])
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Abstract

The d-dimensional Hamming torus is the graph whose vertices are all of the integer points inside an a1n × a2n × ⋅⋅⋅ × adn box in $\mathbb{R}^d$ (for constants a1, . . ., ad > 0), and whose edges connect all vertices within Hamming distance one. We study the size of the largest connected component of the subgraph generated by independently removing each vertex of the Hamming torus with probability 1 − p. We show that if p = λ/n, then there exists λc > 0, which is the positive root of a degree d polynomial whose coefficients depend on a1, . . ., ad, such that for λ < λc the largest component has O(log n) vertices (w.h.p. as n → ∞), and for λ > λc the largest component has $(1-q) \lambda \bigl(\prod_i a_i \bigr) n^{d-1} + o (n^{d-1})$ vertices and the second largest component has O(log n) vertices w.h.p. An implicit formula for q < 1 is also given. The value of λc that we find is distinct from the critical value for the emergence of a giant component in bond percolation on the Hamming torus.

Keywords

Primary 05C80Secondary 82B43
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 2 , March 2014 , pp. 290 - 315
Copyright
Copyright © Cambridge University Press 2013 

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