Slightly subcritical hypercube percolation

Tim Hulshof*, Asaf Nachmias

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)557-593
Number of pages37
JournalRandom Structures and Algorithms
Issue number2
StatePublished - 1 Mar 2020


  • diameter
  • hypercube
  • mixing time
  • percolation
  • subcriticality


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