Reconstruction for Colorings on Trees
Nayantara Bhatnagar, Juan Vera, Eric Vigoda, Dror Weitz

TL;DR
This paper investigates the conditions under which the root of a colored tree remains independent of leaf colorings, establishing thresholds for extremality and implications for Markov chain mixing times.
Contribution
It proves that for certain parameters, the Gibbs measure is strongly extremal, leading to rapid mixing of local Markov chains, extending previous results on colorings of trees.
Findings
Reconstruction is possible when k<Δ/ln(Δ).
The measure is strongly extremal for k=CΔ/ln(Δ) with C>1.
Rapid mixing of local Markov chains is achieved under these conditions.
Abstract
Consider -colorings of the complete tree of depth and branching factor . If we fix the coloring of the leaves, as tends to , for what range of is the root uniformly distributed over all colors? This corresponds to the threshold for uniqueness of the infinite-volume Gibbs measure. It is straightforward to show the existence of colorings of the leaves which ``freeze'' the entire tree when . For , Jonasson proved the root is ``unbiased'' for any fixed coloring of the leaves and thus the Gibbs measure is unique. What happens for a {\em typical} coloring of the leaves? When the leaves have a non-vanishing influence on the root in expectation, over random colorings of the leaves, reconstruction is said to hold. Non-reconstruction is equivalent to extremality of the free-boundary Gibbs measure. When ,…
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