Robust reconstruction on trees is determined by the second eigenvalue

Details Robust reconstruction on trees is determined by the second eigenvalue Robust reconstruction on trees is determined by the second eigenvalue

2004-06-23

arxiv.org/abs/math/0406447v2

Annals of Probability 2004, Vol. 32, No. 3, 2630-2649

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117904000000153 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117904000000153

Consider a Markov chain on an infinite tree T=(V,E) rooted at \rho. In such a chain, once the initial root state \sigma(\rho) 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 \mu_j denote the resulting measure for \sigma(\rho)=j. The resulting measure \mu_j is defined on configurations \sigma=(\sigma(x))_{x\in V}\in A^V, where A is some finite set. Let \mu_j^n denote the restriction of \mu to the sigma-algebra generated 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 is solvable if lim inf_{n\to\infty}\alpha_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 \nu be a nondegenerate distribution on A and \epsilon >0. Let \sigma be chosen according to \mu_j^n and \sigma' be obtained from \sigma by letting for each node independently, \sigma(v)=\sigma'(v) with probability 1-\epsilon and \sigma'(v) be an independent sample from \nu otherwise. We denote by \mu_j^n[\nu,\epsilon ] the resulting measure on \sigma'. The measure \mu_j^n[\nu,\epsilon ] is a perturbation of the measure \mu_j^n.

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