Slightly subcritical hypercube percolation

Tim Hulshof; Asaf Nachmias

doi:10.1002/rsa.20853

Slightly subcritical hypercube percolation

Tim Hulshof, Asaf Nachmias

School of Mathematical Sciences

Tel Aviv University

Research output: Contribution to journal › Article › peer-review

Abstract

We study bond percolation on the hypercube {0,1}^m in the slightly subcritical regime where p = p_c(1 − ε_m) and ε_m = o(1) but ε_m ≫ 2^−m/3 and study the clusters of largest volume and diameter. We establish that with high probability the largest component has cardinality (Formula presented.), that the maximal diameter of all clusters is (Formula presented.), and that the maximal mixing time of all clusters is (Formula presented.). These results hold in different levels of generality, and in particular, some of the estimates hold for various classes of graphs such as high-dimensional tori, expanders of high degree and girth, products of complete graphs, and infinite lattices in high dimensions.

Original language	English
Pages (from-to)	557-593
Number of pages	37
Journal	Random Structures and Algorithms
Volume	56
Issue number	2
DOIs	https://doi.org/10.1002/rsa.20853
State	Published - 1 Mar 2020

Keywords

diameter
hypercube
mixing time
percolation
subcriticality

All Science Journal Classification (ASJC) codes

Software
General Mathematics
Computer Graphics and Computer-Aided Design
Applied Mathematics

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10.1002/rsa.20853

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title = "Slightly subcritical hypercube percolation",

abstract = "We study bond percolation on the hypercube {0,1}m in the slightly subcritical regime where p = pc(1 − εm) and εm = o(1) but εm ≫ 2−m/3 and study the clusters of largest volume and diameter. We establish that with high probability the largest component has cardinality (Formula presented.), that the maximal diameter of all clusters is (Formula presented.), and that the maximal mixing time of all clusters is (Formula presented.). These results hold in different levels of generality, and in particular, some of the estimates hold for various classes of graphs such as high-dimensional tori, expanders of high degree and girth, products of complete graphs, and infinite lattices in high dimensions.",

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AU - Nachmias, Asaf

PY - 2020/3/1

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N2 - We study bond percolation on the hypercube {0,1}m in the slightly subcritical regime where p = pc(1 − εm) and εm = o(1) but εm ≫ 2−m/3 and study the clusters of largest volume and diameter. We establish that with high probability the largest component has cardinality (Formula presented.), that the maximal diameter of all clusters is (Formula presented.), and that the maximal mixing time of all clusters is (Formula presented.). These results hold in different levels of generality, and in particular, some of the estimates hold for various classes of graphs such as high-dimensional tori, expanders of high degree and girth, products of complete graphs, and infinite lattices in high dimensions.

AB - We study bond percolation on the hypercube {0,1}m in the slightly subcritical regime where p = pc(1 − εm) and εm = o(1) but εm ≫ 2−m/3 and study the clusters of largest volume and diameter. We establish that with high probability the largest component has cardinality (Formula presented.), that the maximal diameter of all clusters is (Formula presented.), and that the maximal mixing time of all clusters is (Formula presented.). These results hold in different levels of generality, and in particular, some of the estimates hold for various classes of graphs such as high-dimensional tori, expanders of high degree and girth, products of complete graphs, and infinite lattices in high dimensions.

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KW - hypercube

KW - mixing time

KW - percolation

KW - subcriticality

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M3 - مقالة

SN - 1042-9832

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EP - 593

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

IS - 2

ER -

Slightly subcritical hypercube percolation

Abstract

Keywords

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