  The purpose of this note is twofold. First, we survey the study of the
percolation phase transition on the Hamming hypercube {0,1}^m obtained in the
series of papers [9,10,11,24]. Secondly, we explain how this study can be
performed without the use of the so-called "lace-expansion" technique. To that
aim, we provide a novel simple proof that the triangle condition holds at the
critical probability. We hope that some of these techniques will be useful to
obtain non-perturbative proofs in the analogous, yet much more difficult study
on high-dimensional tori.
