Slightly subcritical hypercube percolation.

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 Θεm−2log(εm32m), that the maximal diameter of all clusters is (1+o(1))εm−1log(εm32m), and that the maximal mixing time of all clusters is Θεm−3log2(εm32m). 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. [ABSTRACT FROM AUTHOR]