  A voter sits on each vertex of an infinite tree of degree $k$, and has to
decide between two alternative opinions. At each time step, each voter switches
to the opinion of the majority of her neighbors. We analyze this majority
process when opinions are initialized to independent and identically
distributed random variables. In particular, we bound the threshold value of
the initial bias such that the process converges to consensus. In order to
prove an upper bound, we characterize the process of a single node in the large
$k$-limit. This approach is inspired by the theory of mean field spin-glass and
can potentially be generalized to a wider class of models. We also derive a
lower bound that is nontrivial for small, odd values of $k$.
