Source: Ann. Appl. Probab. Volume 13, Number 3
(2003), 817-844.
Consider a tree network $T$, where each edge acts as an independent copy of
a given channel $M$, and information is propagated from the root.
For which $T$ and $M$ does the configuration
obtained at level $n$ of $T$
typically contain significant information on the root variable?
This problem arose independently in biology, information theory and
statistical physics.
For all $b$, we construct a channel
for which the variable at the root of the break $b$-ary tree
is independent of the configuration at the second level of that tree,
yet for sufficiently large $B>b$, the mutual information between
the configuration at level $n$ of
the $B$-ary tree and the root variable is bounded away from zero
for all $n$.
This construction is related to Reed--Solomon codes.
We improve the upper bounds on information flow
for asymmetric binary channels (which correspond to the Ising model
with an external field) and for symmetric $q$-ary channels
(which correspond to Potts models).
Let $\lam_2(M)$ denote the second largest
eigenvalue of $M$, in absolute value. A CLT of Kesten and
Stigum implies that if
$b |\lam_2(M)|^2 >1$, then the census
of the variables at any level of
the $b$-ary tree, contains significant information on the root variable.
We establish a converse: If $b |\lam_2(M)|^2 < 1$, then the
census of the variables at level $n$ of the $b$-ary tree is
asymptotically independent of the root variable.
This contrasts with examples where $b |\lam_2(M)|^2 <1$,
yet the configuration at level $n$
is not asymptotically independent of the root variable.
References
[1] ATHREy A, K. B. and NEY, P. E. (1972). Branching Processes. Springer, New York.
[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. Phy s. 79 473-482.
[3] BRIGHTWELL, G. and WINKLER, P. (1999). Graph homomorphisms and phase transitions. J. Combin. Theory Ser. B 77 415-435.
[4] BRIGHTWELL, G. and WINKLER, P. (2000). Gibbs measures and dismantlable graphs, J. Combin. Theory Ser. B 78 141-169.
[5] BRIGHTWELL, G. and WINKLER, P. (2001). Random colorings of a Cay ley tree. Unpublished manuscript.
[6] CAVENDER, J. (1978). Taxonomy with confidence. Math. Biosci. 40 271-280.
[7] COVER, T. M. and THOMAS, J. A. (1991). Elements of Information Theory. Wiley, New York.
[8] DEMBO, A. and ZEITOUNI, O. (1997). Large Deviations, Techniques and Applications. Springer, New York.
[9] DOy LE, P. G. and SNELL, E. J. (1984). Random Walks and Electrical Networks. Math. Assoc. Amer., Washington, DC.
[10] EVANS, W., KENy ON, C., PERES, Y. and SCHULMAN, L. J. (2000). Broadcasting on trees and the Ising model. Ann. Appl. Probab. 10 410-433.
[11] FITCH, W. M. (1971). Toward defining the course of evolution: Minimum change for a specific tree topology. Sy stematic Zoology 20 406-416.
[12] HAJEK, B. and WELLER, T. (1991). On the maximum tolerable noise for reliable computation by formulas. IEEE Trans. Inform. Theory 37 388-391.
[13] HIGUCHI, Y. (1977). Remarks on the limiting Gibbs state on a (d + 1)-tree. Publ. RIMS Ky oto Univ. 13 335-348.
[14] IOFFE, D. (1996). A note on the extremality of the disordered state for the Ising model on the Bethe lattice. Lett. Math. Phy s. 37 137-143.
[15] 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 and A. Roualt, eds.). Birkhäuser, Boston.
[16] KENy ON, C., MOSSEL, E. and PERES, Y. (2001). Glauber dy namics on trees and hy perbolic graphs (extended abstract). In Proccedings of FOCS 2001. To appear.
[17] KESTEN, H. and STIGUM, B. P. (1966). Limit theorems for decomposable multidimensional Galton-Watson processes. J. Math. Anal. Appl. 17 309-338.
[18] KESTEN, H. and STIGUM, B. P. (1966). Additional limit theorem for indecomposable multidimensional Galton-Watson processes. Ann. Math. Statist. 37 1463-1481.
Mathematical Reviews (MathSciNet):
MR200979
[19] LOVÁSZ, L. and WINKLER, P. (1998). Mixing times. DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 41 85-133.
[20] Ly ONS, R. (1990). Random walks and percolation on trees. Ann. Probab. 18 931-958.
[21] Ly ONS, R. and PEMANTLE, R. (1992). Random walk in a random environment and firstpassage percolation on trees. Ann. Probab. 20 125-136.
[22] MOSSEL, E. (1998). Recursive reconstruction on periodic trees. Random Structures Algorithms 13 81-97.
[23] MOSSEL, E. (2001). Reconstruction on trees: Beating the second eigenvalue. Ann. Appl. Probab. 11 285-300.
[24] PEMANTLE, R. and STEIF, J. E. (1999). Robust phase transitions for Heisenberg and other models on general trees. Ann. Probab. 27 876-912.
[25] REED, I. S. and SOLOMON, G. (1954). A class of multiple-error-correcting codes and the decoding scheme. IEEE Trans. Inform. Theory 4 38-49.
[26] SHAMIR, A. (1979). How to share a secret? Comm. ACM 22 612-613.
[27] SPITZER, F. (1975). Markov random fields on an infinite tree. Ann. Probab. 3 387-394.
[28] STEEL, M. (1989). Distributions in bicolored evolutionary trees. Ph.D. thesis, Massey Univ., Palmerston North, New Zealand.
