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 μj denote the resulting measure for σ(ρ) = j. The resulting measure μj is defined on configurations σ = (σ(x))x ∈ V ∈ script A signV, where script A sign is some finite set. Let μjn denote the restriction of μ to the sigma-algebra generated by the variables σ(x), where x is at distance exactly n from ρ. Letting αn = maxi, j ∈ script A sign dTV (μin, μjn), where dTV denotes total variation distance, we say that the reconstruction problem is solvable if lim infn→¿ αn → 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 → ∞. In this paper we study the problem of robust reconstruction. Let ν be a nondegenerate distribution on script A sign and ε > 0. Let σ be chosen according to μjn and σ′ be obtained from σ by letting for each node independently, σ(υ) = σ(υ) with probability 1 - ε and σ′(υ) be an independent sample from ν otherwise. We denote by μjn[ν, ε] the resulting measure on σ′. The measure μjn[ν, ε] is a perturbation of the measure μjn. Letting αn(ν, ε) = maxi, j ∈ script A sign dTV(μin[ν, ε], μjn[ν, ε]), we say that the reconstruction problem is ν-robust-solvable if lim inf n→∞ αn(ν, ε) > 0 for all 0 < ε < 1. Roughly speaking, the reconstruction problem is robust-solvable if for any noise-rate and for all n, the nth level of the tree contains a nonvanishing amount of information on the root of the tree. Standard techniques imply that if T is the rooted B-ary tree (where each node has B children) and if B|λ2(M)|2 > 1, where λ2(M) is the second largest eigenvalue of M (in absolute value), then for all nondegenerate ν, the reconstruction problem is ν-robust-solvable. We prove a converse and show that the reconstruction problem is not ν-robust-solvable if B|λ2(M)|2 < 1. This proves a conjecture by the second author and Y. Peres. We also consider other models of noise and general trees.