Broadcasting on Trees and the Ising ModelWilliam

Consider a process in which information is transmitted from a given root node on a noisy tree network T. We start with an unbiased random bit R at the root of the tree, and send it down the edges of T. On every edge the bit can be reversed with probability , and these errors occur independently. The goal is to reconstruct R from the values which arrive at the nth level of the tree. This model has been studied in information theory, genetics and statistical mechanics. We bound the reconstruction probability from above using the maximum ow on T viewed as a capacitated network, and from below using the electrical conductance of T. For general innnite trees, we establish a sharp threshold: The probability of correct reconstruction tends to 1=2 as n ! 1 if (1 ?2) 2 < p c (T), but the reconstruction probability stays bounded away from 1=2 if the opposite inequality holds. Here p c (T) is the critical probability for percolation on T; in particular p c (T) = 1=b for the b+1-regular tree. The asymptotic reconstruction problem is equivalent to purity of the \free boundary" Gibbs state for the Ising model on a tree. The special case of regular trees was solved in 1995 by Bleher, Ruiz and Zagrebnov; our extension to general trees depends on a coupling argument, and on a reconstruction algorithm that weights the input bits by the electrical current ow from the root to the leaves.

[1]  Graham R. Brightwell,et al.  Graph Homomorphisms and Phase Transitions , 1999, J. Comb. Theory, Ser. B.

[2]  Elchanan Mossel.  Recursive reconstruction on periodic trees , 1998, Random Struct. Algorithms.

[3]  Robin Pemantle,et al.  Unpredictable paths and percolation , 1998 .

[4]  William S. Evans,et al.  On the Maximum Tolerable Noise for Reliable Computation by Formulas , 1998, IEEE Trans. Inf. Theory.

[5]  D. Ioffe.  On the extremality of the disordered state for the Ising model on the Bethe lattice , 1996 .

[6]  L. Schulman,et al.  Information theory and noisy computation , 1995, Proceedings of 1995 IEEE International Symposium on Information Theory.

[7]  J. Ruiz,et al.  On the purity of the limiting gibbs state for the Ising model on the Bethe lattice , 1995 .

[8]  M. Steel,et al.  Five surprising properties of parsimoniously colored trees , 1995 .

[9]  R. Pemantle,et al.  Domination Between Trees and Application to an Explosion Problem , 2004, math/0404044.

[10]  Leonard J. Schulman,et al.  Signal propagation, with application to a lower bound on the depth of noisy formulas , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[11]  Russell Lyons,et al.  Random Walks, Capacity and Percolation on Trees , 1992 .

[12]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[13]  R. Lyons.  Random Walks and Percolation on Trees , 1990 .

[14]  R. Lyons.  The Ising model and percolation on trees and tree-like graphs , 1989 .

[15]  I. Vajda.  Theory of statistical inference and information , 1989 .

[16]  Nicholas Pippenger,et al.  Reliable computation by formulas in the presence of noise , 1988, IEEE Trans. Inf. Theory.

[17]  J. Laurie Snell,et al.  Random Walks and Electrical Networks , 1984 .

[18]  J. Snell,et al.  A branching process showing a phase transition , 1979, Journal of Applied Probability.

[19]  Y. Higuchi.  Remarks on the Limiting GIbbs States on a (d+1)-Tree , 1977 .

[20]  C. Preston.  Gibbs States on Countable Sets , 1974 .

[21]  H. Kesten,et al.  Additional Limit Theorems for Indecomposable Multidimensional Galton-Watson Processes , 1966 .

[22]  J. von Neumann,et al.  Probabilistic Logic and the Synthesis of Reliable Organisms from Unreliable Components , 1956 .