The Glauber Dynamics for Colourings of Bounded Degree Trees

Lucier, Brendan; Molloy, Michael; Peres, Yuval

doi:10.1007/978-3-642-03685-9_47

Cited by 15 publications

(22 citation statements)

References 17 publications

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“…Our result extends to more general k and b, thereby refining the general picture provided by [12] and [21]. …”

Section: Introductionsupporting

confidence: 80%

“…Our main result provides (nearly) sharp bounds on the mixing time and relaxation time of the Glauber dynamics for the complete tree, establishing a phase transition at the critical point k = b(1 + o b (1))/ ln b. Our proofs build upon the approaches used by [12] and [21]. Theorem 1.1.…”

Section: Introductionmentioning

confidence: 80%

“…Thus we want to establish a more precise picture than provided by the results of [12] and [21]. Our main result provides (nearly) sharp bounds on the mixing time and relaxation time of the Glauber dynamics for the complete tree, establishing a phase transition at the critical point k = b(1 + o b (1))/ ln b.…”

Section: Introductionmentioning

confidence: 87%

“…Let τ * be the relaxation time of the Glauber dynamics on the star graph G * using k colors. We use the following decomposition result of Lucier and Molloy [21], which is an application of the block dynamics technique (see, Proposition 3.4 in [23]) to the Glauber dynamics on the complete trees combined with Lemma 2 in Mossel and Sly [27].…”

Section: Upper Boundsmentioning

confidence: 99%

“…Subsequently, improved results were established for the tree. In particular, Goldberg et al [12] proved the mixing time is n Ω(b/(k ln b)) for the complete tree with branching factor b, and Lucier et al [21,22] proved the mixing time is n O(1+b/(k ln b)) for any tree with maximum degree b.…”

Section: Introductionmentioning

confidence: 99%

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Phase Transition for the Mixing Time of the Glauber Dynamics for Coloring Regular Trees

Tetali¹,

Vera²,

Vigoda³

et al. 2010

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms

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We prove that the mixing time of the Glauber dynamics for random k-colorings of the complete tree with branching factor b undergoes a phase transition at k = b(1+o b (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+o b (1) ln 2 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) andThe critical point C = 1 is interesting since it coincides (at least up to first order) to the so-called reconstruction threshold which was recently established by Sly. The reconstruction threshold has been of considerable interest recently since it appears to have close connections to the efficiency of certain local algorithms, and this work was inspired by our attempt to understand these connections in this particular setting.

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“…Our result extends to more general k and b, thereby refining the general picture provided by [12] and [21]. …”

Section: Introductionsupporting

confidence: 80%

Section: Introductionmentioning

confidence: 80%

Section: Introductionmentioning

confidence: 87%

Section: Upper Boundsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Phase Transition for the Mixing Time of the Glauber Dynamics for Coloring Regular Trees

Tetali¹,

Vera²,

Vigoda³

et al. 2010

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms

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Randomly coloring planar graphs with fewer colors than the maximum degree

Hayes

Vera

Vigoda

2014

Random Struct. Alg.

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We study Markov chains for randomly sampling k‐colorings of a graph with maximum degree Δ. Our main result is a polynomial upper bound on the mixing time of the single‐site update chain known as the Glauber dynamics for planar graphs when k=Ω(Δ/logΔ). Our results can be partially extended to the more general case where the maximum eigenvalue of the adjacency matrix of the graph is at most Δ1−ϵ, for fixed ϵ>0. The main challenge when k≤Δ+1 is the possibility of “frozen” vertices, that is, vertices for which only one color is possible, conditioned on the colors of its neighbors. Indeed, when Δ=O(1), even a typical coloring can have a constant fraction of the vertices frozen. Our proofs rely on recent advances in techniques for bounding mixing time using “local uniformity” properties. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 731–759, 2015

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The Glauber dynamics for edge‐colorings of trees

Delcourt

Heinrich

Perarnau

2020

Random Struct Algorithms

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Let T be a tree on n vertices and with maximum degree Δ. We show that for k ≥ Δ + 1 the Glauber dynamics for k-edgecolorings of T mixes in polynomial time in n. The bound on the number of colors is best possible as the chain is not even ergodic for k ≤ Δ. Our proof uses a recursive decomposition of the tree into subtrees; we bound the relaxation time of the original tree in terms of the relaxation time of its subtrees using block dynamics and chain comparison techniques. Of independent interest, we also introduce a monotonicity result for Glauber dynamics that simplifies our proof.

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The Glauber Dynamics for Colourings of Bounded Degree Trees

Cited by 15 publications

References 17 publications

Phase Transition for the Mixing Time of the Glauber Dynamics for Coloring Regular Trees

Phase Transition for the Mixing Time of the Glauber Dynamics for Coloring Regular Trees

Randomly coloring planar graphs with fewer colors than the maximum degree

The Glauber dynamics for edge‐colorings of trees

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