The Kesten-Stigum Reconstruction Bound is Tight for Roughly Symmetric Binary Channels

We establish the exact threshold for the reconstruction problem for a binary asymmetric channel on the b-ary tree, provided that the asymmetry is sufficiently small. This is the first exact reconstruction threshold obtained in roughly a decade. We discuss the implications of our result for Glauber dynamics, phylogenetic reconstruction, noisy communication and the so-called “replica symmetry breaking” in spin glasses and random satisfiability problems.