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 $\sigma=(\sigma(x))_{x\in V}\in \mathcal {A}^{V}$, where $\mathcal {A}$ is some finite set. Let μ_jⁿ 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 $\alpha_{n}=\max_{i,j\in \mathcal {A}}d_{\mathrm{TV}}(\mu_{i}^{n},\mu_{j}^{n})$, where d_TV denotes total variation distance, we say that the reconstruction problem is solvable if lim inf _n→∞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→∞.

In this paper we study the problem of robust reconstruction. Let ν be a nondegenerate distribution on $\mathcal {A}$" and ɛ>0. Let σ be chosen according to μ_jⁿ and σ' be obtained from σ by letting for each node independently, σ(v)=σ'(v) with probability 1−ɛ and σ'(v) be an independent sample from ν otherwise. We denote by μ_jⁿ[ν,ɛ] the resulting measure on σ'. The measure μ_jⁿ[ν,ɛ] is a perturbation of the measure μ_jⁿ. Letting $\alpha_{n}(\nu,\varepsilon )=\max_{i,j\in \mathcal {A}}d_{\mathrm{TV}}(\mu_{i}^{n}[\nu,\varepsilon ],\mu_{j}^{n}[\nu,\varepsilon ])$, we say that the reconstruction problem is ν-robust-solvable if lim inf _n→∞α_n(ν,ɛ)>0 for all 0<ɛ<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.