Percolation on irregular high-dimensional product graphs

Sahar Diskin*, Joshua Erde, Mihyun Kang, Michael Krivelevich

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We consider bond percolation on high-dimensional product graphs G = □ i=1t G(i), where □ denotes the Cartesian product. We call the G(i) the base graphs and the product graph G the host graph. Very recently, Lichev (J. Graph Theory, 99(4):651-670, 2022) showed that, under a mild requirement on the isoperimetric properties of the base graphs, the component structure of the percolated graph Gp undergoes a phase transition when p is around 1|d, where d is the average degree of the host graph. In the supercritical regime, we strengthen Lichev's result by showing that the giant component is in fact unique, with all other components of order o(|G|), and determining the sharp asymptotic order of the giant. Furthermore, we answer two questions posed by Lichev (J. Graph Theory, 99(4):651-670, 2022): firstly, we provide a construction showing that the requirement of bounded degree is necessary for the likely emergence of a linear order component; secondly, we show that the isoperimetric requirement on the base graphs can be, in fact, super-exponentially small in the dimension. Finally, in the subcritical regime, we give an example showing that in the case of irregular high-dimensional product graphs, there can be a polynomially large component with high probability, very much unlike the quantitative behaviour seen in the Erdos-Rényi random graph and in the percolated hypercube, and in fact in any regular high-dimensional product graphs, as shown by the authors in a companion paper (Percolation on high-dimensional product graphs. arXiv:2209.03722, 2022).

Original languageEnglish
JournalCombinatorics Probability and Computing
DOIs
StateAccepted/In press - 2023

Funding

FundersFunder number
USA-Israel BSF2018267
Austrian Science FundI6502, W1230, P36131

    Keywords

    • Bond percolation
    • component sizes
    • random subgraphs

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