Reconstruction on Trees: Beating the Second Eigenvalue
Statistics and Probability
We consider a process in which information is transmitted from a given root node on a noisy d-ary tree network T. We start with a uniform symbol taken from an alphabet A. Each edge of the tree is an independent copy of some channel (Markov chain) M, where M is irreducible and aperiodic on A. The goal is to reconstruct the symbol at the root from the symbols at the nth level of the tree. This model has been studied in information theory, genetics and statistical physics. The basic question is: is it possible to reconstruct (some information on)the root? In other words, does the probability of correct reconstruction tend to 1/A as n →∞? It is known that reconstruction is possible if dλ22(M) > 1, where λ2(M) is the second eigenvalue of M. Moreover, in this case it is possible to reconstruct using a majority algorithm which ignores the location of the data at the boundary of the tree. When M is a symmetric binary channel, this threshold is sharp. In this paper we show that, both for the binary asymmetric channel and for the symmetric channel on many symbols, it is sometimes possible to reconstruct even when dλ22(M) < 1. This result indicates that, for many (maybe most) tree-indexed Markov chains, the location of the data on the boundary plays a crucial role in reconstruction problems.