## Statistics Papers

#### Document Type

Journal Article

#### Date of this Version

2004

#### Publication Source

Annals of Probability

#### Volume

32

#### Issue

3B

#### Start Page

2630

#### Last Page

2649

#### DOI

10.1214/009117904000000153

#### 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 , where is some finite set. Let μ_{j}^{n} 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 , 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 *n*th 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 and ɛ>0. Let σ be chosen according to μ_{j}^{n} 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_{j}^{n}[ν,ɛ] the resulting measure on σ'. The measure μ_{j}^{n}[ν,ɛ] is a perturbation of the measure μ_{j}^{n}. Letting , 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 *n*th 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*|λ_{2}(*M*)|^{2}>1, where λ_{2}(*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*|λ_{2}(*M*)|^{2}<1. This proves a conjecture by the second author and Y. Peres. We also consider other models of noise and general trees.

#### Keywords

robust phase transition, reconstruction on trees, branching number

#### Recommended Citation

Janson, S.,
&
Mossel, E.
(2004).
Robust Reconstruction on Trees is Determined by the Second Eigenvalue.
*Annals of Probability,*
*32*
(3B),
2630-2649.
http://dx.doi.org/10.1214/009117904000000153

**Date Posted:** 27 November 2017

This document has been peer reviewed.