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Reconstruction for colorings on trees
, 2008
"... Consider kcolorings of the complete tree of depth ℓ and branching factor ∆. If we fix the coloring of the leaves, for what range of k is the root uniformly distributed over all k colors (in the limit ℓ → ∞)? This corresponds to the threshold for uniqueness of the infinitevolume Gibbs measure. It i ..."
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Cited by 11 (4 self)
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Consider kcolorings of the complete tree of depth ℓ and branching factor ∆. If we fix the coloring of the leaves, for what range of k is the root uniformly distributed over all k colors (in the limit ℓ → ∞)? This corresponds to the threshold for uniqueness of the infinitevolume Gibbs measure. It is straightforward to show the existence of colorings of the leaves which “freeze ” the entire tree when k ≤ ∆ + 1. For k ≥ ∆ + 2, Jonasson proved the root is “unbiased ” for any fixed coloring of the leaves and thus the Gibbs measure is unique. What happens for a typical coloring of the leaves? When the leaves have a nonvanishing influence on the root in expectation, over random colorings of the leaves, reconstruction is said to hold. Nonreconstruction is equivalent to extremality of the Gibbs measure. When k < ∆ / ln ∆, it is straightforward to show that reconstruction is possible (and hence the measure is not extremal). We prove that for C> 2 and k = C∆ / ln ∆, nonreconstruction holds, i.e., the Gibbs measure is extremal. We prove a strong form of extremality: with high probability over the colorings of the leaves the influence at the root decays exponentially fast with the depth of the tree. These are the first results coming close to the threshold for extremality for colorings. Extremality on trees and random graphs has received considerable attention recently since it may have connections to the efficiency of local algorithms.
Phase Transition for the Mixing Time of the Glauber Dynamics for Coloring Regular Trees
"... We prove that the mixing time of the Glauber dynamics for random kcolorings of the complete tree with branching factor b undergoes a phase transition at k = b(1+ob(1)) / ln b. Our main result shows nearly sharp bounds on the mixing time of the dynamics on the complete tree with n vertices for k = C ..."
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Cited by 6 (4 self)
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We prove that the mixing time of the Glauber dynamics for random kcolorings of the complete tree with branching factor b undergoes a phase transition at k = b(1+ob(1)) / ln b. Our main result shows nearly sharp bounds on the mixing time of the dynamics on the complete tree with n vertices for k = Cb / ln b colors with constant C. For C ≥ 1 we prove the mixing time is O(n 1+ob(1) 2 ln n). On the other side, for C < 1 the mixing time experiences a slowing down, in particular, we prove it is O(n 1/C+o b(1) ln 2 n) and Ω(n 1/C−o b(1)). The critical point C = 1 is interesting
The Glauber dynamics for colourings of bounded degree trees
, 2008
"... We study the Glauber dynamics Markov chain for kcolourings of trees with maximum degree ∆. For k ≥ 3, we show that the mixing time on every tree is at most n O(1+∆/(k log ∆)). This bound is tight up to the constant factor in the exponent, as evidenced by the complete tree. Our proof uses a weighted ..."
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We study the Glauber dynamics Markov chain for kcolourings of trees with maximum degree ∆. For k ≥ 3, we show that the mixing time on every tree is at most n O(1+∆/(k log ∆)). This bound is tight up to the constant factor in the exponent, as evidenced by the complete tree. Our proof uses a weighted canonical paths analysis and a variation of the block dynamics in which we exploit the differing relaxation times of blocks.