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 Θ(ε _m ⁻² log(ε _m ³ 2 ^m )), that the maximal diameter of all clusters is (1+o(1))ε _m ⁻¹ log(ε _m ³ 2 ^m ), and that the maximal mixing time of all clusters is Θ(ε _m ⁻³ log ² (ε _m ³ 2 ^m )). 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.