Robust reconstruction on trees is determined by the second eigenvalue

In this paper we study the perturbative theory of reconstruction on trees, and show how it depends on the spectrum of the underlying Markov chain. In particular, we show that the threshold for ``robust reconstruction'' for the $B$-ary tree is $B |\lam_2(M)|^2 = 1$, where $\lam_2(M)$ denotes the eigenvalue of $M$ which is the second largest in absolute value. We prove a similar threshold for general bounded degree trees, where $B$ is replaced by the branching number of the tree $\br(T)$.