The Annals of Applied Probability

Reconstruction on Trees: Beating the Second Eigenvalue

Elchanan Mossel

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Abstract

We consider a process in which information is transmitted from a given root node on a noisy -dary tree network T. We start with a uniform symbol taken from an alphabet \(\mathcal{A}\). Each edge of the tree is an independent copy of some channel (Markov chain) M, where M is irreducible and aperiodic on \(\mathcal{A}\). 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 \(1 /{\mathcal{A}}\) 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.

Article information

Source
Ann. Appl. Probab. Volume 11, Number 1 (2001), 285-300.

Dates
First available in Project Euclid: 27 August 2001

Permanent link to this document
http://projecteuclid.org/euclid.aoap/998926994

Digital Object Identifier
doi:10.1214/aoap/998926994

Mathematical Reviews number (MathSciNet)
MR1825467

Zentralblatt MATH identifier
1021.90008

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 90B15: Network models, stochastic 68R99: None of the above, but in this section

Keywords
tree Markov chain percolation Ising model Potts model coupling

Citation

Mossel, Elchanan. Reconstruction on Trees: Beating the Second Eigenvalue. Ann. Appl. Probab. 11 (2001), no. 1, 285--300. doi:10.1214/aoap/998926994. http://projecteuclid.org/euclid.aoap/998926994.


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References

  • [1] Atherya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, Berlin.
  • [2] Bleher, P. M., Ruiz, J. and Zagrebnov V. A. (1995). On the purity of limiting Gibbs state for the Ising model on the Bethe lattice. J. Statist. Phys. 79 473-482.
  • [3] Cover, T. M. and Thomas, J. A. (1991). Elements of Information Theory. Wiley, New York.
  • [4] Evans, W., Kenyon, C., Peres, Y. and Schulman L. J. (2000). Broadcasting on trees and the Ising model. Ann. Appl. Probab. 10 410-433.
  • [5] Kesten, H. and Stigum, B. P. (1966). Additional limit theorem for indecomposable multidimensional Galton-Watson processes. Ann. Math. Statist. 37 1463-1481.
  • [6] Lyons, R. (1990). Random walks and percolation on trees. Ann. Probab. 18 931-958.
  • [7] Mossel, E. (1998). Recursive reconstruction on periodic trees. Random Structures Algorithms 13 81-97.
  • [8] Mossel, E. and Peres, Y. (2000). Coupling for tree-indexed Markov chains. In preparation.