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Reconstruction thresholds on regular trees | | James B. Martin
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28 May 2003 | | Subject: | Probability MSC-class: 60K35 | math.PR | | Abstract: | We consider a branching random walk with binary state space and index set $T^k$, the infinite rooted tree in which each node has k children (also known as the model of "broadcasting on a tree"). 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 non-vanishing information about the value at the root as $d oinfty$. Adapting 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 $p_{11}=0$. 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 (k+1)-regular tree with activity $lambda=1$, the unique simple invariant Gibbs measure is extremal in the set of Gibbs measures, for any k. | | Source: | arXiv, math.PR/0305400 | | Services: | Forum | Review | PDF | Favorites | |
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