## The Annals of Applied Probability

### Broadcasting on trees and the Ising model

#### Abstract

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.

#### Article information

Source
Ann. Appl. Probab. Volume 10, Number 2 (2000), 410-433.

Dates
First available in Project Euclid: 22 April 2002

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1019487349

Digital Object Identifier
doi:10.1214/aoap/1019487349

Mathematical Reviews number (MathSciNet)
MR1768240

Zentralblatt MATH identifier
1052.60076

#### Citation

Evans, William; Kenyon, Claire; Peres, Yuval; Schulman, Leonard J. Broadcasting on trees and the Ising model. Ann. Appl. Probab. 10 (2000), no. 2, 410--433. doi:10.1214/aoap/1019487349. http://projecteuclid.org/euclid.aoap/1019487349.

#### References

• [1] Benjamini, I., Pemantle, R. and Peres, Y. (1998). Unpredictable paths and percolation. Ann. Probab. 26 1198-1211.
• [2] 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.
• [3] Brightwell, G. R. and Winkler, P. (1999). Graph homomorphisms and phase transitions. J. Combin. Theory Ser. A. To appear.
• [4] Cavender, J. (1978). Taxonomy with confidence. Math. Biosci. 40 271-280.
• [5] 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.
• [6] Cover, T. M. and Thomas, J. A. (1991). Elements of Information Theory. Wiley, New York.
• [7] Doyle, P. G. and Snell, E. J. (1984). Random Walks and Electrical Networks. Math. Assoc. Amer., Washington, D.C.
• [8] Evans, W. (1994). Information theory and noisy computation. Ph.D. dissertation, Dept. Computer Science, Univ. California, Berkeley.
• [9] Evans, W., Kenyon, C., Peres, Y. and Schulman, L. J. (1995). A critical phenomenon in a broadcast process. Unpublished manuscript.
• [10] 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.
• [11] Evans, W. and Schulman, L. J. (1994). Lower bound on the depth of noisy circuits. Unpublished manuscript.
• [12] 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
• [13] Georgii, H. O. (1988). Gibbs Measures and Phase Transitions. de Gruyter, Berlin.
• [14] 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.
• [15] 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.
• [16] Hajek, B. and Weller, T. (1991). On the maximum tolerable noise for reliable computation by formulas. IEEE Trans. Inform. Theory 37 388-391.
• [17] Hartigan, J. A. (1971). Minimum mutation fits to a given tree. Biometrics 29 53-65.
• [18] Higuchi, Y. (1977). Remarks on the limiting Gibbs state on a d+1 -tree. Publ. RIMS Kyoto Univ. 13 335-348.
• [19] 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.
• [20] 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.
• [21] Kesten, H. and Stigum, B. P. (1966). Additional limit theorems for indecomposable multidimensional Galton-Watson processes. Ann. Math. Statist. 37 1463-1481.
• [22] Le Cam, L. (1974). Notes on Asymptotic Methods in Statistical Decision Theory. Centre de Rech. Math., Univ. Montr´eal.
• [23] Lyons, R. (1989). The Ising model and percolation on trees and tree-like graphs. Comm. Math. Phys. 125 337-352.
• [24] Lyons, R. (1990). Random walks and percolation on trees. Ann. Probab. 18 931-958.
• [25] Lyons, R. (1992). Random walks, capacity, and percolation on trees. Ann. Probab. 20 2043- 2088.
• [26] Lyons, R. and Peres, Y. (1997). Probability on Trees and Networks. To appear. Available at http://php.indiana.edu/ rdlyons/.
• [27] Moore, T. and Snell, J. L. (1979). A branching process showing a phase transition. J. Appl. Probab. 16 252-260.
• [28] Mossel, E. (1998). Recursive reconstruction on periodic trees. Random Structures Algorithms 13 81-97.
• [29] Mossel, E. (1999). Reconstruction on trees: beating the second eigenvalue. Unpublished manuscript.
• [30] Pemantle, R. and Peres, Y. (1994). Domination between trees and application to an explosion problem. Ann. Probab. 22 180-194.
• [31] Pemantle, R. and Peres, Y. (1995). Recursions on trees and the Ising model at critical temperatures. Unpublished manuscript.
• [32] Pippenger, N. (1988). Reliable computation by formulas in the presence of noise. IEEE Trans. Inform. Theory 34 194-197.
• [33] Preston, C. J. (1974). Gibbs States on Countable Sets. Cambridge Univ. Press.
• [34] Spitzer, F. (1975). Markov random fields on an infinite tree. Ann Probab. 3 387-394.
• [35] Steel, M. (1989). Distribution in bicolored evolutionary trees. Ph.D. thesis, Massey Univ., Palmerston North, New Zealand.
• [36] Steel, M. and Charleston, M. (1995). Five surprising properties of parsimoniously colored trees. Bull. Math. Biology 57 367-375.
• [37] Vajda, I. (1989). Theory of Statistical Inference and Information. Kluwer, Dordrecht.
• [38] 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