  In this paper, we consider a broadcasting process in which information is
propagated from a given root node on a noisy tree network, and answer the
question that whether the symbols at the nth level of the tree contain
non-vanishing information of the root as n goes to infinity. Although the
reconstruction problem on the tree has been studied in numerous contexts
including information theory, mathematical genetics and statistical physics,
the existing literatures with rigorous reconstruction thresholds established
are very limited. In the remarkable work of Borgs, Chayes, Mossel and Roch (The
Kesten-Stigum reconstruction bound is tight for roughly symmetric binary
channels), the exact threshold for the reconstruction problem for a binary
asymmetric channel on the d-ary tree is establish, provided that the asymmetry
is sufficiently small, which is the first exact reconstruction threshold
obtained in roughly a decade. In this paper, by means of refined analyses of
moment recursion on a weighted version of the magnetization, and concentration
investigations, we rigorously give a complete answer to the question of how
small it needs to be to establish the tightness of the reconstruction threshold
and further determine its asymptotics of large degrees.
