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10
2007 A survey on the use of Markov chains to randomly sample colourings
 In Combinatorics, Complexity and Chance (eds G Grimmett, C McDiarmid
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Randomly Coloring Graphs of Girth at Least Five (Extended Abstract)
 STOC'03
, 2003
"... We improve rapid mixing results for the simple Glauber dynamics designed to generate a random kcoloring of a boundeddegree graph. ..."
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Cited by 18 (3 self)
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We improve rapid mixing results for the simple Glauber dynamics designed to generate a random kcoloring of a boundeddegree graph.
Approximate counting via correlation decay in spin systems
 In Proceedings of the 23rd Annual ACMSIAM Symposium on Discrete Algorithms
, 2012
"... We give the first deterministic fully polynomialtime approximation scheme (FPTAS) for computing the partition function of a twostate spin system on an arbitrary graph, when the parameters of the system satisfy the uniqueness condition on infinite regular trees. This condition is of physical signif ..."
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Cited by 16 (10 self)
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We give the first deterministic fully polynomialtime approximation scheme (FPTAS) for computing the partition function of a twostate spin system on an arbitrary graph, when the parameters of the system satisfy the uniqueness condition on infinite regular trees. This condition is of physical significance and is believed to be the right boundary between approximable and inapproximable. The FPTAS is based on the correlation decay technique introduced by Bandyopadhyay and Gamarnik [1] and Weitz [61]. The classic correlation decay is defined with respect to graph distance. Although this definition has natural physical meanings, it does not directly support an FPTAS for systems on arbitrary graphs, because for graphs with unbounded degrees, the local computation that provides a desirable precision by correlation decay may take superpolynomial time. We introduce a notion of computationally efficient correlation decay, in which the correlation decay is measured in a refined metric instead of graph distance. We use a potential method to analyze the amortized behavior of this correlation decay and establish a correlation decay that guarantees an inversepolynomial precision by polynomialtime local computation. This gives us an FPTAS for spin systems on arbitrary graphs. This new notion of correlation decay properly reflects the algorithmic aspect of the spin systems, and may be used for designing FPTAS for other counting problems. 1
Sampling grid colourings with fewer colours
 PROC. OF LATIN ’04
, 2004
"... We provide an optimally mixing Markov chain for 6colourings of the square grid. Furthermore, this implies that the uniform distribution on the set of such colourings has strong spatial mixing. 4 and 5 are now the only remaining values of k for which it is not known whether there exists a rapidly mi ..."
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Cited by 6 (0 self)
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We provide an optimally mixing Markov chain for 6colourings of the square grid. Furthermore, this implies that the uniform distribution on the set of such colourings has strong spatial mixing. 4 and 5 are now the only remaining values of k for which it is not known whether there exists a rapidly mixing Markov chain for kcolourings of the square grid.
Variable Length Path Coupling
, 2006
"... We present a new technique for constructing and analyzing couplings to bound the convergence rate of finite Markov chains. Our main theorem is a generalization of the path coupling theorem of Bubley and Dyer, allowing the defining partial couplings to have length determined by a random stopping time ..."
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Cited by 5 (2 self)
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We present a new technique for constructing and analyzing couplings to bound the convergence rate of finite Markov chains. Our main theorem is a generalization of the path coupling theorem of Bubley and Dyer, allowing the defining partial couplings to have length determined by a random stopping time. Unlike the original path coupling theorem, our version can produce multistep (nonMarkovian) couplings. Using our variable length path coupling theorem, we improve the upper bound on the mixing time of the Glauber dynamics for randomly sampling colorings.
CONCENTRATION OF MEASURE AND MIXING FOR MARKOV CHAINS
, 2008
"... We consider Markovian models on graphs with local dynamics. We show that, under suitable conditions, such Markov chains exhibit both rapid convergence to equilibrium and strong concentration of measure in the stationary distribution. We illustrate our results with applications to some known chains ..."
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Cited by 3 (1 self)
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We consider Markovian models on graphs with local dynamics. We show that, under suitable conditions, such Markov chains exhibit both rapid convergence to equilibrium and strong concentration of measure in the stationary distribution. We illustrate our results with applications to some known chains from computer science and statistical mechanics.
Sampling 3colourings of regular bipartite graphs
, 2006
"... We show that if Σ = (V,E) is a regular bipartite graph for which the expansion of subsets of a single parity of V is reasonably good and which satisfies a certain local condition (that the union of the neighbourhoods of adjacent vertices does not contain too many pairwise nonadjacent vertices), a ..."
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We show that if Σ = (V,E) is a regular bipartite graph for which the expansion of subsets of a single parity of V is reasonably good and which satisfies a certain local condition (that the union of the neighbourhoods of adjacent vertices does not contain too many pairwise nonadjacent vertices), and if M is a Markov chain on the set of proper 3colourings of Σ which updates the colour of at most ρV  vertices at each step and whose stationary distribution is uniform, then for ρ ≈.22 and d sufficiently large the convergence to stationarity ofM is (essentially) exponential in V . In particular, if Σ is the ddimensional hypercube Qd (the graph on vertex set {0, 1}d in which two strings are adjacent if they differ on exactly one coordinate) then the convergence to stationarity of the wellknown Glauber (singlesite update) dynamics is exponentially slow in
The Glauber dynamics on colourings of a graph with high girth and maximum degree
 In Proc. of the 34th ACM Symposium on Theory of Computing (STOC
, 2001
"... We prove that the Glauber dynamics on the C colourings of a graph G on n vertices with girth g and maximum degree mixes rapidly if (i) C = q and q > q where q = 1:4890::: is the root of = 1; and (ii) D log n and g D log for some constant D = D(q). This improves the corresponding ..."
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We prove that the Glauber dynamics on the C colourings of a graph G on n vertices with girth g and maximum degree mixes rapidly if (i) C = q and q > q where q = 1:4890::: is the root of = 1; and (ii) D log n and g D log for some constant D = D(q). This improves the corresponding result with q 1:763 obtained by Dyer and Frieze [FOCS01] for the same class of graphs. It also improves q 11=6 1:833 obtained by Vigoda [FOCS99] for general graphs.
Faster mixing and . . .
 PROBAB. THEORY RELAT. FIELDS 137, 475–486 (2007)
, 2007
"... We prove a new bound on the mixing time of a Markov chain by considering the conductance of its connected subsets. ..."
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We prove a new bound on the mixing time of a Markov chain by considering the conductance of its connected subsets.