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Consider a Markov chain on an infinite tree T=(V,E) rooted at \rho. In such achain, once the initial root state \sigma(\rho) is chosen, each vertexiteratively chooses its state from the one of its parent by an application of aMarkov transition rule (and all such applications are independent). Let \mu_jdenote the resulting measure for \sigma(\rho)=j. The resulting measure \mu_j isdefined on configurations \sigma=(\sigma(x))_{x\in V}\in A^V, where A is somefinite set. Let \mu_j^n denote the restriction of \mu to the sigma-algebragenerated by the variables \sigma(x), where x is at distance exactly n from\rho. Letting \alpha_n=max_{i,j\in A}d_{TV}(\mu_i^n,\mu_j^n), where d_{TV}denotes total variation distance, we say that the reconstruction problem issolvable if lim inf_{n\to\infty}\alpha_n>0. Reconstruction solvability roughlymeans that the nth level of the tree contains a nonvanishing amount ofinformation on the root of the tree as n\to\infty. In this paper we study theproblem of robust reconstruction. Let \nu be a nondegenerate distribution on Aand \epsilon >0. Let \sigma be chosen according to \mu_j^n and \sigma' beobtained from \sigma by letting for each node independently,\sigma(v)=\sigma'(v) with probability 1-\epsilon and \sigma'(v) be anindependent sample from \nu otherwise. We denote by \mu_j^n[\nu,\epsilon ] theresulting measure on \sigma'. The measure \mu_j^n[\nu,\epsilon ] is aperturbation of the measure \mu_j^n.

Robust reconstruction on trees is determined by the second eigenvalue