Robust reconstruction on trees is determined by the second eigenvalue

Abstract

Consider a Markov chain on an infinite tree T=(V,E) rooted at ρ. In such a chain, once the initial root state σ(ρ) is chosen, each vertex iteratively chooses its state from the one of its parent by an application of a Markov transition rule (and all such applications are independent). Let μ_j denote the resulting measure for σ(ρ)=j. The resulting measure μ_j is defined on configurations σ=(σ(x))_{x\in V}\in A^V, where A is some finite set. Let μ_j^n denote the restriction of μto the sigma-algebra generated by the variables σ(x), where x is at distance exactly n from ρ. Letting α_n=max_{i,j\in A}d_{TV}(μ_i^n,μ_j^n), where d_{TV} denotes total variation distance, we say that the reconstruction problem is solvable if lim inf_{n\to\infty}α_n>0. Reconstruction solvability roughly means that the nth level of the tree contains a nonvanishing amount of information on the root of the tree as n\to\infty. In this paper we study the problem of robust reconstruction. Let νbe a nondegenerate distribution on A and ε>0. Let σbe chosen according to μ_j^n and σ' be obtained from σby letting for each node independently, σ(v)=σ'(v) with probability 1-εand σ'(v) be an independent sample from νotherwise. We denote by μ_j^n[ν,ε] the resulting measure on σ'. The measure μ_j^n[ν,ε] is a perturbation of the measure μ_j^n.