Source: Ann. Appl. Probab.
Volume 10, Number 2
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 $\varepsilon$, and these errors occur
independently. The goal is to reconstruct $R$ from the values which arrive at
the $n$th 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 flow on $T$ viewed as a capacitated
network, and from below using the electrical conductance of $T$. For general
infinite trees, we establish a sharp threshold: the probability of correct
reconstruction tends to 1/2 as $n \to \infty$ if $(1 - 2\varepsilon)^2 <
p_c(T)$, but the reconstruction probability stays bounded away from ½ 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 algorighm that weights the input bits by the electrical current
flow from the root to the leaves.
 Benjamini, I., Pemantle, R. and Peres, Y. (1998). Unpredictable paths and percolation. Ann. Probab. 26 1198-1211.
 Bleher, P. M., Ruiz, J. and Zagrebnov, V. A. (1995). On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice. J. Statist. Phys. 79 473-482.
 Brightwell, G. R. and Winkler, P. (1999). Graph homomorphisms and phase transitions. J. Combin. Theory Ser. A. To appear.
 Cavender, J. (1978). Taxonomy with confidence. Math. Biosci. 40 271-280.
 Chayes, J. T., Chayes, L., Sethna, J. P. and Thouless, D. J. (1986). A mean field spin glass with short range interactions, Comm. Math. Phys. 106 41-89.
 Cover, T. M. and Thomas, J. A. (1991). Elements of Information Theory. Wiley, New York.
 Doyle, P. G. and Snell, E. J. (1984). Random Walks and Electrical Networks. Math. Assoc. Amer., Washington, D.C.
 Evans, W. (1994). Information theory and noisy computation. Ph.D. dissertation, Dept. Computer Science, Univ. California, Berkeley.
 Evans, W., Kenyon, C., Peres, Y. and Schulman, L. J. (1995). A critical phenomenon in a broadcast process. Unpublished manuscript.
 Evans, W. and Schulman, L. J. (1993). Signal propagation, with application to a lower bound on the depth of noisy formulas. In Proceedings of the 34th Annual Symposium on Foundations of Computer Science 594-603.
 Evans, W. and Schulman, L. J. (1994). Lower bound on the depth of noisy circuits. Unpublished manuscript.
 Fitch, W. M. (1971). Toward defining the course of evolution: minimum change for a specific tree topology. Syst. Zool. 20 406-416.
432 EVANS, KENYON, PERES AND SCHULMAN
 Georgii, H. O. (1988). Gibbs Measures and Phase Transitions. de Gruyter, Berlin.
 Grimmett, G. R. (1996). Percolation and disordered systems. Lectures in Probability Theory and Statistics, Ecole d'Et´e de Probabilit´es de Saint Flour XXVI. Lecture Notes in Math. 1665 153-300. Springer, Berlin.
 H¨aggstr ¨om, O. and Mossel, E. (1998). A nearest neighbor process with low predictability and percolation in 2 + dimensions. Ann. Probab. 26 1212-1231.
 Hajek, B. and Weller, T. (1991). On the maximum tolerable noise for reliable computation by formulas. IEEE Trans. Inform. Theory 37 388-391.
 Hartigan, J. A. (1971). Minimum mutation fits to a given tree. Biometrics 29 53-65.
 Higuchi, Y. (1977). Remarks on the limiting Gibbs state on a d+1 -tree. Publ. RIMS Kyoto Univ. 13 335-348.
 Ioffe, D. (1996). A note on the extremality of the disordered state for the Ising model on the Bethe lattice. Lett. Math. Phys. 37 137-143.
 Ioffe, D. (1996). A note on the extremality of the disordered state for the Ising model on the Bethe lattice. In Trees (B. Chauvin, S. Cohen, A. Roualt, eds.). Birkh¨auser, Boston.
 Kesten, H. and Stigum, B. P. (1966). Additional limit theorems for indecomposable multidimensional Galton-Watson processes. Ann. Math. Statist. 37 1463-1481.
Mathematical Reviews (MathSciNet): MR200979
 Le Cam, L. (1974). Notes on Asymptotic Methods in Statistical Decision Theory. Centre de Rech. Math., Univ. Montr´eal.
 Lyons, R. (1989). The Ising model and percolation on trees and tree-like graphs. Comm. Math. Phys. 125 337-352.
 Lyons, R. (1990). Random walks and percolation on trees. Ann. Probab. 18 931-958.
 Lyons, R. (1992). Random walks, capacity, and percolation on trees. Ann. Probab. 20 2043- 2088.
 Lyons, R. and Peres, Y. (1997). Probability on Trees and Networks. To appear. Available at http://php.indiana.edu/ rdlyons/.
 Moore, T. and Snell, J. L. (1979). A branching process showing a phase transition. J. Appl. Probab. 16 252-260.
Mathematical Reviews (MathSciNet): MR531760
 Mossel, E. (1998). Recursive reconstruction on periodic trees. Random Structures Algorithms 13 81-97.
 Mossel, E. (1999). Reconstruction on trees: beating the second eigenvalue. Unpublished manuscript.
 Pemantle, R. and Peres, Y. (1994). Domination between trees and application to an explosion problem. Ann. Probab. 22 180-194.
 Pemantle, R. and Peres, Y. (1995). Recursions on trees and the Ising model at critical temperatures. Unpublished manuscript.
 Pippenger, N. (1988). Reliable computation by formulas in the presence of noise. IEEE Trans. Inform. Theory 34 194-197.
Mathematical Reviews (MathSciNet): MR945312
 Preston, C. J. (1974). Gibbs States on Countable Sets. Cambridge Univ. Press.
 Spitzer, F. (1975). Markov random fields on an infinite tree. Ann Probab. 3 387-394.
 Steel, M. (1989). Distribution in bicolored evolutionary trees. Ph.D. thesis, Massey Univ., Palmerston North, New Zealand.
 Steel, M. and Charleston, M. (1995). Five surprising properties of parsimoniously colored trees. Bull. Math. Biology 57 367-375.
 Vajda, I. (1989). Theory of Statistical Inference and Information. Kluwer, Dordrecht.
 von Neumann, J. (1956). Probabilistic logics and the synthesis of reliable organisms from unreliable components. In Automata Studies (C. E. Shannon and J. McCarthy, eds.) 43-98. Princeton Univ. Press.
LRI, Universit´e de Paris-Sud Orsay France E-mail: Claire.Kenyon@lri.fr