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 ε, 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 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 → ∞ if 1 − 2ε 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 flow from the root to the leaves.

Phase ordering after a deep quench: the stochastic Ising and hard core gas models on a tree

“…Another way to look at β 1 is to say that µ free is an extremal Gibbs measure iff β β 1 (see [4,15,16,2] and, more recently, [24]). Finally β 1 has also the interpretation of the non-reconstruction/reconstruction threshold in the context of "bit reconstruction problems" on a noisy symmetric channel [10,27,26].…”

The Ising model on trees: boundary conditions and mixing time

“…Remark: The validity of ÎÅ, i.e, the decay of point-to-set correlations, is of interest independently of its implication for the spectral gap (an implication which is new to this paper): e.g., it is closely related to the purity of the infinite volume Gibbs measure and to bit reconstruction problems on trees [12]. In the special case of a free boundary and ¼ , part´ µ of Theorem 4.1 was first proved in [7] via a lengthy calculation, which was considerably simplified in [19].…”

“…It was later reproved in [4] (for arbitrary boundary conditions) as a consequence of the fact that the spectral gap is bounded in this situation. An extension to general trees can be found in [12] and [20]. Our motivation for presenting another proof of part´ µ (in addition to handling general fields ) is the simplicity of our argument compared with previous ones.…”

“…I.e., there is now an "intermediate" region ¬ ¼ ¬ ¬ ½ in which the´·µ-and´ µ-boundaries exert long-range influence but typical boundaries do not, while in the "low temperature" region ¬ ¬ ½ long-range influence occurs even for typical boundaries. ¬ ½ has alternative interpretations as the critical value for extremality of the Gibbs measure and the threshold for noisy data transmission on the tree [12]. The Glauber dynamics for the Ising model on trees has also been studied.…”

“…We stress that, while the tree is simpler in some respects than due to the lack of cycles, in other respects it is more complex: e.g., it exhibits a "double phase transition" (see below). Moreover, the Ising model on trees has recently received a lot of attention as the canonical example of a statistical physics model on a "non-amenable" graph (i.e., one whose boundary is of comparable size to its volume) -see, e.g., [4,6,7,12,19,24,27,39]. In the next subsection, we briefly describe the Ising model on trees before stating our results in more detail.…”