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...ph. Previously, other Markov chains for graph colourings were only known to be rapidly mixing for k ≥ 2∆, including the simple Markov chain discovered independently by Jerrum [17] and Salas and Sokal =-=[27]-=-. Using the comparison technique described in this section, we can conclude that the Jerrum/Salas–Sokal chain is also rapidly mixing for k = 5 when G has maximum degree 3, and for k = 7 when G is a tr...

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... valuable to understand the asymptotic behavior of the chromatic roots of Km,n as m, n → ∞ in various ways (e.g. with α = m/n fixed). Using the Dobrushin uniqueness theorem [45, 99], it can be proven =-=[87]-=- that for a countable graph G of maximum degree r, the q-state Potts-model Gibbs measure on 11 Biggs, Damerell and Sands [12] have verified that the chromatic roots of all 3-regular graphs with ≤ 10 v...

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...perature (at least for Q ? 2\Delta), by Sokal [21]. The latter result follows directly from the standard proof of the Dobrushin uniqueness theorem [4, 5], combined with Salas and Sokal's verification =-=[20]-=- of the hypotheses of that theorem for the case of the antiferromagnetic Potts model with Q ? 2\Delta. 2.2 The random cluster model Before introducing the Swendsen-Wang approach for approximating the ...

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...hat the couplings used are optimal in a sense which we define. 1 Introduction Markov chains on the set of proper colourings of graphs have been studied in computer science [9] and statistical physics =-=[13]-=-. In both applications, the rapidity of convergence of the chain is the main focus of interest, though for somewhat different reasons. The papers [9, 13] introduced a simple Markov chain, which we sha...

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...o these previous methods do not apply. 4 The Bounding Chain for k-colorings Our bounding chain will give information about a particle moving according to the heat bath Markov chain of Salas and Sokal =-=[11]-=- that has as its state space the proper k-colorings of a graph G. In one step of the heat bath chain, we choose a vertex uniformly at random and choose a color for the vertex uniformly at random from ...

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...an approximation algorithm with complexity 2O(log 2 n). Namely, our algorithm is super-polynomial but still significantly quicker than exponential time. Although our regime α > 2.8432 . . . is weaker than q/∆ > 2, for which the Markov chain is known to mix 1245 rapidly, our method has advantages over the Markov chains based method, the most notable of which is the deterministic nature of the algorithm, which eliminates the sampling error. Our approach is based on establishing a certain correlation decay property which has been considered in many settings [BW04], including the coloring problem [SS97], [GMP05], [BW02],[Jon02] and has been recently a subject of interest. In particular, the correlation decay has been established in [GMP05] for coloring triangle-free graphs under the assumption that α > α∗ = 1.763 . . ., the unique solution of αe− 1 α = 1. (Some mild additional assumptions were adopted). The principal motivation for establishing correlation decay property comes from statistical physics, in particular the connection between the correlation decay and the uniqueness of the associated Gibbs measure (uniform measure in our setting) on infinite versions of the graph, typically latt...

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...e show uniqueness here is still larger than that given by the original Dobrushin condition, which is q > 2(max {b1, b2} + 1). (The calculation using the original Dobrushin condition was first done in =-=[18]-=-, and implies uniqueness for any underlying graph and q > 2 maxx deg(x), where deg(x) stands for the number of neighbors of x). In order to use our theorems we need to specify a coupled update rule fo...