Discrete Random Walks, DRW'03
Cyril Banderier and Christian Krattenthaler (eds.)
DMTCS Conference Volume AC (2003), pp. 191204
author:  James B. Martin 

title:  Reconstruction Thresholds on Regular Trees 
keywords:  broadcasting on a tree, reconstruction, hardcore model, Gibbs measure, extremality 
abstract: 
We consider the model of broadcasting on a tree, with
binary state space, on the infinite rooted tree
T
in which each node has
k
k
children. The root of the tree takes a random value 0
or 1, and then each node passes a value independently to
each of its children according to a 2x2 transition matrix
P
. We say that reconstruction is possible if the
values at the
d
th level of the tree contain nonvanishing
information about the value at the root as
d→∞
. Extending a method of Brightwell and Winkler, we
obtain new conditions under which reconstruction is
impossible, both in the general case and in the special
case
p
. The latter case is closely related to the hardcore
model from statistical physics; a corollary of our results
is that, for the hardcore model on the
11
=0
(k+1)
regular tree with activity
λ=1
, the unique simple invariant Gibbs measure is
extremal in the set of Gibbs measures, for any
k ≥ 2
.

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reference:  James B. Martin (2003), Reconstruction Thresholds on Regular Trees, in Discrete Random Walks, DRW'03, Cyril Banderier and Christian Krattenthaler (eds.), Discrete Mathematics and Theoretical Computer Science Proceedings AC, pp. 191204 
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