Results 1 
4 of
4
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 ..."
Abstract

Cited by 12 (5 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
(Show Context)
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
Phase Transition for Glauber Dynamics for Independent Sets on Regular Trees
"... We study the effect of boundary conditions on the relaxation time of the Glauber dynamics for the hardcore lattice gas model on the nvertex regular bary tree of height h. The hardcore model is defined on independent sets weighted by an activity (or fugacity) λ on trees. Reconstruction studies the ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
We study the effect of boundary conditions on the relaxation time of the Glauber dynamics for the hardcore lattice gas model on the nvertex regular bary tree of height h. The hardcore model is defined on independent sets weighted by an activity (or fugacity) λ on trees. Reconstruction studies the effect of a ‘typical’ boundary condition, i.e., fixed assignment to the leaves, on the root. The threshold for when reconstruction occurs (and a typical boundary influences the root in the limit h → ∞) has been of considerable recent interest since it appears to be connected to the efficiency of certain local algorithms on locally treelike graphs. The reconstruction threshold occurs at ω ≈ ln b/b where λ = ω(1 + ω) b is a convenient reparameterization of the model. We prove that for all boundary conditions, the relaxation time τ in the nonreconstruction region is fast, namely τ = O ( n 1+ob(1) ) for any ω ≤ ln b/b. In the reconstruction region, for all boundary conditions, we prove τ = O ( n 1+δ+ob(1) ) for ω = (1+δ) ln b/b, for every δ> 0. In contrast, we construct a boundary condition, for which the Glauber dynamics slows down in the reconstruction region, namely τ = Ω ( n1+δ/2−ob(1) ) for ω = (1 + δ) ln b/b, for every δ> 0. The interesting part of our proof is this lower bound result, which uses a general technique that transforms an algorithm to prove reconstruction into a set in the state space of the Glauber dynamics with poor conductance.