The mixing time of Glauber dynamics for coloring regular trees

2010 ◽  
pp. NA-NA
Author(s):  
Leslie Ann Goldberg ◽  
Mark Jerrum ◽  
Marek Karpinski
Keyword(s):  
Mixing Time ◽  
Glauber Dynamics

Frozen (Δ + 1)-colourings of bounded degree graphs

2020 ◽  
pp. 1-14
Author(s):  
Marthe Bonamy ◽  
Nicolas Bousquet ◽  
Guillem Perarnau
Keyword(s):  
Graph Theory ◽  
Maximum Degree ◽  
Mixing Time ◽  
Glauber Dynamics ◽  
Regular Graphs ◽  
Connected Component ◽  
Bounded Degree ◽  
Large Girth ◽  
Bounded Degree Graphs

Abstract Let G be a graph on n vertices and with maximum degree Δ, and let k be an integer. The k-recolouring graph of G is the graph whose vertices are k-colourings of G and where two k-colourings are adjacent if they differ at exactly one vertex. It is well known that the k-recolouring graph is connected for $k\geq \Delta+2$ . Feghali, Johnson and Paulusma (J. Graph Theory83 (2016) 340–358) showed that the (Δ + 1)-recolouring graph is composed by a unique connected component of size at least 2 and (possibly many) isolated vertices. In this paper, we study the proportion of isolated vertices (also called frozen colourings) in the (Δ + 1)-recolouring graph. Our first contribution is to show that if G is connected, the proportion of frozen colourings of G is exponentially smaller in n than the total number of colourings. This motivates the use of the Glauber dynamics to approximate the number of (Δ + 1)-colourings of a graph. In contrast to the conjectured mixing time of O(nlog n) for $k\geq \Delta+2$ colours, we show that the mixing time of the Glauber dynamics for (Δ + 1)-colourings restricted to non-frozen colourings can be Ω(n2). Finally, we prove some results about the existence of graphs with large girth and frozen colourings, and study frozen colourings in random regular graphs.

scholarly journals Phase transition for the mixing time of the Glauber dynamics for coloring regular trees

2012 ◽  
Vol 22 (6) ◽  
pp. 2210-2239 ◽  
Author(s):  
Prasad Tetali ◽  
Juan C. Vera ◽  
Eric Vigoda ◽  
Linji Yang
Keyword(s):  
Phase Transition ◽  
Mixing Time ◽  
Glauber Dynamics

scholarly journals A note on perfect simulation for Exponential Random Graph Models

2020 ◽  
Vol 24 ◽  
pp. 138-147 ◽  
Author(s):  
Andressa Cerqueira ◽  
Aurélien Garivier ◽  
Florencia Leonardi
Keyword(s):  
Markov Chain ◽  
Random Graph ◽  
Mixing Time ◽  
Graph Model ◽  
Glauber Dynamics ◽  
Perfect Simulation ◽  
Exponential Random Graph Models ◽  
Graph Models ◽  
Random Graph Model ◽  
Exponential Random Graph

In this paper, we propose a perfect simulation algorithm for the Exponential Random Graph Model, based on the Coupling from the past method of Propp and Wilson (1996). We use a Glauber dynamics to construct the Markov Chain and we prove the monotonicity of the ERGM for a subset of the parametric space. We also obtain an upper bound on the running time of the algorithm that depends on the mixing time of the Markov chain.

scholarly journals The Mixing Time Evolution of Glauber Dynamics for the Mean-Field Ising Model

2009 ◽  
Vol 289 (2) ◽  
pp. 725-764 ◽  
Author(s):  
Jian Ding ◽  
Eyal Lubetzky ◽  
Yuval Peres
Keyword(s):  
Ising Model ◽  
Time Evolution ◽  
Mixing Time ◽  
Mean Field ◽  
Glauber Dynamics ◽  
The Mean

scholarly journals Random-Cluster Dynamics on Random Regular Graphs in Tree Uniqueness

2021 ◽  
Author(s):  
Antonio Blanca ◽  
Reza Gheissari
Keyword(s):  
Phase Transition ◽  
Potts Model ◽  
Cluster Model ◽  
Iterative Scheme ◽  
Mixing Time ◽  
Glauber Dynamics ◽  
Regular Graphs ◽  
Random Cluster Model ◽  
Regular Tree ◽  
Random Cluster

AbstractWe establish rapid mixing of the random-cluster Glauber dynamics on random $$\varDelta $$ Δ -regular graphs for all $$q\ge 1$$ q ≥ 1 and $$p<p_u(q,\varDelta )$$ p < p u ( q , Δ ) , where the threshold $$p_u(q,\varDelta )$$ p u ( q , Δ ) corresponds to a uniqueness/non-uniqueness phase transition for the random-cluster model on the (infinite) $$\varDelta $$ Δ -regular tree. It is expected that this threshold is sharp, and for $$q>2$$ q > 2 the Glauber dynamics on random $$\varDelta $$ Δ -regular graphs undergoes an exponential slowdown at $$p_u(q,\varDelta )$$ p u ( q , Δ ) . More precisely, we show that for every $$q\ge 1$$ q ≥ 1 , $$\varDelta \ge 3$$ Δ ≥ 3 , and $$p<p_u(q,\varDelta )$$ p < p u ( q , Δ ) , with probability $$1-o(1)$$ 1 - o ( 1 ) over the choice of a random $$\varDelta $$ Δ -regular graph on n vertices, the Glauber dynamics for the random-cluster model has $$\varTheta (n \log n)$$ Θ ( n log n ) mixing time. As a corollary, we deduce fast mixing of the Swendsen–Wang dynamics for the Potts model on random $$\varDelta $$ Δ -regular graphs for every $$q\ge 2$$ q ≥ 2 , in the tree uniqueness region. Our proof relies on a sharp bound on the “shattering time”, i.e., the number of steps required to break up any configuration into $$O(\log n)$$ O ( log n ) sized clusters. This is established by analyzing a delicate and novel iterative scheme to simultaneously reveal the underlying random graph with clusters of the Glauber dynamics configuration on it, at a given time.

scholarly journals A Note on the Glauber Dynamics for Sampling Independent Sets

10.37236/1552 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Eric Vigoda
Keyword(s):  
Maximum Degree ◽  
Mixing Time ◽  
Independent Set ◽  
Glauber Dynamics ◽  
Independent Sets ◽  
General Graphs

This note considers the problem of sampling from the set of weighted independent sets of a graph with maximum degree $\Delta$. For a positive fugacity $\lambda$, the weight of an independent set $\sigma$ is $\lambda^{|\sigma|}$. Luby and Vigoda proved that the Glauber dynamics, which only changes the configuration at a randomly chosen vertex in each step, has mixing time $O(n\log{n})$ when $\lambda < {{2}\over {\Delta-2}}$ for triangle-free graphs. We extend their approach to general graphs.

Dobrushin Conditions and Systematic Scan

2008 ◽  
Vol 17 (6) ◽  
pp. 761-779 ◽  
Author(s):  
MARTIN DYER ◽  
LESLIE ANN GOLDBERG ◽  
MARK JERRUM
Keyword(s):  
Mixing Time ◽  
Heat Bath ◽  
Glauber Dynamics ◽  
Rapid Mixing ◽  
Path Coupling ◽  
Total Influence ◽  
Dobrushin Uniqueness ◽  
A Site ◽  
Matrix Balancing ◽  
The Relationship

We consider Glauber dynamics on finite spin systems. The mixing time of Glauber dynamics can be bounded in terms of the influences of sites on each other. We consider three parameters bounding these influences: α, the total influence on a site, as studied by Dobrushin; α′, the total influence of a site, as studied by Dobrushin and Shlosman; and α″, the total influence of a site in any given context, which is related to the path-coupling method of Bubley and Dyer. It is known that if any of these parameters is less than 1 then random-update Glauber dynamics (in which a randomly chosen site is updated at each step) is rapidly mixing. It is also known that the Dobrushin condition α < 1 implies that systematic-scan Glauber dynamics (in which sites are updated in a deterministic order) is rapidly mixing. This paper studies two related issues, primarily in the context of systematic scan: (1) the relationship between the parameters α, α′ and α″, and (2) the relationship between proofs of rapid mixing using Dobrushin uniqueness (which typically use analysis techniques) and proofs of rapid mixing using path coupling. We use matrix balancing to show that the Dobrushin–Shlosman condition α′ < 1 implies rapid mixing of systematic scan. An interesting question is whether the rapid mixing results for scan can be extended to the α = 1 or α′ = 1 case. We give positive results for the rapid mixing of systematic scan for certain α = 1 cases. As an application, we show rapid mixing of systematic scan (for any scan order) for heat-bath Glauber dynamics for properq-colourings of a degree-Δ graphGwhenq≥ 2Δ.

scholarly journals Glauber Dynamics on Trees: Boundary Conditions and Mixing Time

2004 ◽  
Vol 250 (2) ◽  
pp. 301-334 ◽  
Author(s):  
Fabio Martinelli ◽  
Alistair Sinclair ◽  
Dror Weitz
Keyword(s):  
Boundary Conditions ◽  
Mixing Time ◽  
Glauber Dynamics
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