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Slow mixing of Glauber Dynamics for the hard-core model on regular bipartite graphs [article]

David Galvin, Prasad Tetali
2012 arXiv   pre-print
Let =(V,E) be a finite, d-regular bipartite graph.  ...  For any λ>0 let π_λ be the probability measure on the independent sets of in which the set I is chosen with probability proportional to λ^|I| (π_λ is the hard-core measure with activity λ on ).  ...  The authors thank Microsoft Research for this support.  ... 
arXiv:1206.3165v1 fatcat:yz7j34bnl5b7hkohb6quy7u6ae

Slow mixing of Glauber dynamics for the hard-core model on regular bipartite graphs

David Galvin, Prasad Tetali
2006 Random structures & algorithms (Print)  
Let Σ = (V, E) be a finite, d-regular bipartite graph.  ...  For any λ > 0 let π λ be the probability measure on the independent sets of Σ in which the set I is chosen with probability proportional to λ |I| (π λ is the hard-core measure with activity λ on Σ).  ...  This work originated while the second author was visiting the Theory group at Microsoft Research in Redmond, Washington. The authors thank Microsoft Research for this support.  ... 
doi:10.1002/rsa.20094 fatcat:jr7wok37vbghbaork7vtsggaom

Fast algorithms at low temperatures via Markov chains

Zongchen Chen, Andreas Galanis, Leslie A. Goldberg, Will Perkins, James Stewart, Eric Vigoda
2020 Random structures & algorithms (Print)  
Combining our results for the hard-core and Potts models with Markov chain comparison tools, we obtain polynomial mixing time for Glauber dynamics restricted to appropriate portions of the state space.  ...  Instead, recent work of Jenssen, Keevash, and Perkins yields polynomial-time algorithms in the low-temperature regime on bounded-degree (bipartite) expander graphs using polymer models and the cluster  ...  The extra factor of n in the running time of the sampling algorithm for the hard-core model as compared to the Potts model is due to the fact that the hard-core model on a bipartite graph does not in general  ... 
doi:10.1002/rsa.20968 fatcat:jmvrxgibkfg2hpehmhjyk23wh4

A Personal List of Unsolved Problems Concerning Lattice Gases and Antiferromagnetic Potts Models [article]

Alan D. Sokal
2000 arXiv   pre-print
For each model, I consider its equilibrium properties (uniqueness/nonuniqueness of the infinite-volume Gibbs measure, complex zeros of the partition function) and the dynamics of local and nonlocal Monte  ...  I review recent results and unsolved problems concerning the hard-core lattice gas and the q-coloring model (antiferromagnetic Potts model at zero temperature).  ...  hospitality of John Cardy and the Department of Theoretical Physics.  ... 
arXiv:cond-mat/0004231v3 fatcat:nsopxdq5svdfnkydy7z6u2ugmu

Fast algorithms at low temperatures via Markov chains [article]

Zongchen Chen, Andreas Galanis, Leslie Ann Goldberg, Will Perkins, James Stewart, Eric Vigoda
2021 arXiv   pre-print
for the Potts model and O(n^2 log n) for the hard-core model, in contrast to typical running times of n^O(logΔ) for algorithms based on Barvinok's polynomial interpolation method on graphs of maximum  ...  We apply this Markov chain to polymer models derived from the hard-core and ferromagnetic Potts models on bounded-degree (bipartite) expander graphs.  ...  The extra factor in the running time of the sampling algorithm for the hard-core model as compared to the Potts model is due to the fact that the hard-core model on a bipartite graph does not in general  ... 
arXiv:1901.06653v6 fatcat:3ntpu2kujrb4fjyhyixadtmr4m

Algorithms for #BIS-hard problems on expander graphs [chapter]

Matthew Jenssen, Peter Keevash, Will Perkins
2019 Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms  
The value λ c (∆) is the uniqueness threshold of the hard-core model on the infinite ∆-regular tree [22] .  ...  We give an FPTAS and an efficient sampling algorithm for the high-fugacity hard-core model on bounded-degree bipartite expander graphs and the low-temperature ferromagnetic Potts model on bounded-degree  ...  For λ > λ c (∆), the Glauber dynamics for the hard-core model on G bip n,∆ are known to mix slowly [26] .  ... 
doi:10.1137/1.9781611975482.135 dblp:conf/soda/JenssenKP19 fatcat:nvvcwyd7d5hj3ashc27iphf4aa

Algorithms for #BIS-Hard Problems on Expander Graphs

Matthew Jenssen, Peter Keevash, Will Perkins
2020 SIAM journal on computing (Print)  
Using contour-based techniques Galvin and Tetali [17] showed slow mixing of the Glauber dynamics on ∆-regular bipartite expander graphs.  ...  Similarly, for the hard-core model on a bipartite expander graph with symmetry between the sides of the bipartition, one could follow the suggestion of [21] and start the Glauber dynamics in either the  ... 
doi:10.1137/19m1286669 fatcat:cvn3rdusmncibiansjubwpz434

Counting Colorings of a Regular Graph

David Galvin
2014 Graphs and Combinatorics  
Combin. 13 (2006), #R72, arXiv:1206.3200 • Slow mixing of Glauber dynamics for the hard-core model on regular bipartite graphs (with P.  ...  ACM-IEEE IPSN (2006), 19-26 • Slow mixing of Glauber dynamics for the hard-core model on the hypercube (with P. Tetali), Proc.  ... 
doi:10.1007/s00373-013-1403-z fatcat:qxets5cmk5cp5ijzvn3u66qjme

Ferromagnetic Potts Model: Refined #BIS-hardness and Related Results

Andreas Galanis, Daniel Štefankovič, Eric Vigoda, Linji Yang
2016 SIAM journal on computing (Print)  
Recent results establish for the hard-core model (and more generally for 2-spin antiferromagnetic systems) that the computational complexity of approximating the partition function on graphs of maximum  ...  The #BIS-hardness result uses random bipartite regular graphs as a gadget in the reduction.  ...  There are few results establishing rapid mixing of the Swendsen-Wang algorithm beyond what is known for the Glauber dynamics, see [37] for recent progress showing rapid mixing on the 2-dimensional lattice  ... 
doi:10.1137/140997580 fatcat:5slbs2x5o5gxvfa44ln3bv2ari

Phase Coexistence and Slow Mixing for the Hard-Core Model on ℤ2 [chapter]

Antonio Blanca, David Galvin, Dana Randall, Prasad Tetali
2013 Lecture Notes in Computer Science  
On finite graphs we are interested in determining the mixing time of local Markov chains.  ...  The best result for rapid mixing of local Markov chains on boxes of Z 2 is also when λ < 2.3882 [24] .  ...  Our present work builds on a novel idea from [23] in which the notion of fault lines was introduced to establish slow mixing for the Glauber dynamics on hard-core configurations for moderately large  ... 
doi:10.1007/978-3-642-40328-6_27 fatcat:t3w7uz5sojgktiu74aezbwggv4

Phase Coexistence and Slow Mixing for the Hard-Core Model on Z^2 [article]

Antonio Blanca, David Galvin, Dana Randall, Prasad Tetali
2012 arXiv   pre-print
In the hard-core model on a finite graph we are given a parameter lambda>0, and an independent set I arises with probability proportional to lambda^|I|.  ...  On finite graphs we are interested in determining the mixing time of local Markov chains.  ...  Our present work builds on a novel idea from [23] in which the notion of fault lines was introduced to establish slow mixing for the Glauber dynamics on hard-core configurations for moderately large  ... 
arXiv:1211.6182v1 fatcat:ollqixz7z5hc5eicjeho5fdkaa

Sampling 3-colourings of regular bipartite graphs [article]

David Galvin
2012 arXiv   pre-print
We show that if =(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  ...  the well-known Glauber (single-site update) dynamics is exponentially slow in 2^d/(√(d) d).  ...  We produce the set U by appealing to a lemma from [14] where a similar approximation scheme was used to show that the mixing time of Glauber dynamics for the hard-core model on Q d with activity λ is  ... 
arXiv:1206.3202v1 fatcat:ox4dmxwvk5d4rmepmqh5x2xks4

Sampling 3-colourings of regular bipartite graphs

David Galvin
2007 Electronic Journal of Probability  
of the well-known Glauber (single-site update) dynamics is exponentially slow in 2 d /( √ d log d).  ...  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  ...  We produce the set U by appealing to a lemma from [14] where a similar approximation scheme was used to show that the mixing time of Glauber dynamics for the hard-core model on Q d with activity λ is  ... 
doi:10.1214/ejp.v12-403 fatcat:m2tgfehuhjb3ppb5zznkiuhk3m

Improved Mixing Condition on the Grid for Counting and Sampling Independent Sets [article]

Ricardo Restrepo, Jinwoo Shin, Prasad Tetali, Eric Vigoda, Linji Yang
2011 arXiv   pre-print
Our results imply a fully-polynomial deterministic approximation algorithm for estimating the partition function, as well as rapid mixing of the associated Glauber dynamics to sample from the hard-core  ...  More concretely, let λ_c(T_D) denote the critical value for the so-called uniqueness threshold of the hard-core model on the infinite D-regular tree; recent breakthrough results of Dror Weitz (2006) and  ...  (A related result of Randall [29] showing slow mixing of the Glauber dynamics for λ > 8.066 gives hope for a better upper bound on λ c (Z 2 ).)  ... 
arXiv:1105.0914v3 fatcat:kj3d5sahbvapxi34k2cqgyagsq

Computational Counting (Dagstuhl Seminar 18341)

Ivona Bezáková, Leslie Ann Goldberg, Mark R. Jerrum, Marc Herbstritt
2018 Dagstuhl Reports  
The seminar was held from 20th to 25th August 2017, at Schloss Dagstuhl -Leibnitz Center for Informatics.  ...  This report documents the program and the outcomes of Dagstuhl Seminar 17341 "Computational Counting".  ...  In particular, this yields an approximation for the partition function of the continuous hard core model on a regular graph with large girth in the case λ = 1.  ... 
doi:10.4230/dagrep.7.8.23 dblp:journals/dagstuhl-reports/BezakovaGJ17 fatcat:yp3oqvgo4fal5lbio5yeqt4uje
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