Percolation on 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 percolation on high-dimensional product graphs, where the base graphs are regular and of bounded order. In the subcritical regime, we show that typically the largest component is of order logarithmic in the number of vertices. In the supercritical regime, our main result recovers the sharp asymptotic of the order of the largest component and shows that all the other components are typically of order logarithmic in the number of vertices. In particular, we show that this phase transition is quantitatively similar to the one of the binomial random graph. This generalizes the results of Ajtai, Komlós, and Szemerédi [1] and of Bollobás, Kohayakawa, and Łuczak [5] who showed that the (Formula presented.) -dimensional hypercube, which is the (Formula presented.) -fold Cartesian product of an edge, undergoes a phase transition quantitatively similar to the one of the binomial random graph.

Original languageEnglish
Article numbere21268
JournalRandom Structures and Algorithms
Volume66
Issue number1
DOIs
StatePublished - Jan 2025

Funding

FundersFunder number
Austrian Science Fund
USA–Israel BSF2018267

    Keywords

    • percolation
    • phase transition
    • product graphs
    • random graphs

    Fingerprint

    Dive into the research topics of 'Percolation on High-Dimensional Product Graphs'. Together they form a unique fingerprint.

    Cite this