Summary: Broadcasting on Trees and the Ising Model \Lambda
William Evans Claire Kenyon Yuval Peres y Leonard J. Schulman z
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 ffl, 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 !1 if (1 \Gamma 2ffl) 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.
AMS 1991 subject classifications: Primary 60K35; Secondary 90B15, 68R99.