DMTCS Proceedings, Discrete Random Walks, DRW'03

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DMTCS Conference vol AC (2003), pp. 191-204


Discrete Random Walks, DRW'03

Cyril Banderier and Christian Krattenthaler (eds.)

DMTCS Conference Volume AC (2003), pp. 191-204

author: James B. Martin
title: Reconstruction Thresholds on Regular Trees
keywords: broadcasting on a tree, reconstruction, hard-core model, Gibbs measure, extremality
abstract: We consider the model of broadcasting on a tree, with binary state space, on the infinite rooted tree
in which each node has
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
. We say that reconstruction is possible if the values at the
th level of the tree contain non-vanishing information about the value at the root as
. 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
. The latter case is closely related to the hard-core model from statistical physics; a corollary of our results is that, for the hard-core model on the
-regular tree with activity
, 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. 191-204
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