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, and so-called ``replica symmetry breaking'' in spin glasses and random satisfiability problems. 
Christian Borgs, Jennifer Chayes, Elchanan Mossel, Sebastien Roch