Slightly subcritical hypercube percolation

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.