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RANDOM

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

We consider spin systems on general n-vertex graphs of unbounded degree and explore the effects of spectral independence on the rate of convergence to equilibrium of global Markov chains. Spectral independence is a novel way of quantifying the decay of correlations in spin system models, which has significantly advanced the study of Markov chains for spin systems. We prove that whenever spectral independence holds, the popular Swendsen-Wang dynamics for the q-state ferromagnetic Potts model on graphs of maximum degree Δ, where Δ is allowed to grow with n, converges in O((Δ log n)^c) steps where c > 0 is a constant independent of Δ and n. We also show a similar mixing time bound for the block dynamics of general spin systems, again assuming that spectral independence holds. Finally, for monotone spin systems such as the Ising model and the hardcore model on bipartite graphs, we show that spectral independence implies that the mixing time of the systematic scan dynamics is O(Δ^c log n) for a constant c > 0 independent of Δ and n. Systematic scan dynamics are widely popular but are notoriously difficult to analyze. This result implies optimal O(log n) mixing time bounds for any systematic scan dynamics of the ferromagnetic Ising model on general graphs up to the tree uniqueness threshold. Our main technical contribution is an improved factorization of the entropy functional: this is the common starting point for all our proofs. Specifically, we establish the so-called k-partite factorization of entropy with a constant that depends polynomially on the maximum degree of the graph.

Antonio Blanca and Xusheng Zhang. Rapid Mixing of Global Markov Chains via Spectral Independence: The Unbounded Degree Case. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 53:1-53:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{blanca_et_al:LIPIcs.APPROX/RANDOM.2023.53, author = {Blanca, Antonio and Zhang, Xusheng}, title = {{Rapid Mixing of Global Markov Chains via Spectral Independence: The Unbounded Degree Case}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {53:1--53:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.53}, URN = {urn:nbn:de:0030-drops-188789}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.53}, annote = {Keywords: Markov chains, Spectral Independence, Potts model, Mixing Time} }

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RANDOM

**Published in:** LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

We give an efficient perfect sampling algorithm for weighted, connected induced subgraphs (or graphlets) of rooted, bounded degree graphs. Our algorithm utilizes a vertex-percolation process with a carefully chosen rejection filter and works under a percolation subcriticality condition. We show that this condition is optimal in the sense that the task of (approximately) sampling weighted rooted graphlets becomes impossible in finite expected time for infinite graphs and intractable for finite graphs when the condition does not hold. We apply our sampling algorithm as a subroutine to give near linear-time perfect sampling algorithms for polymer models and weighted non-rooted graphlets in finite graphs, two widely studied yet very different problems. This new perfect sampling algorithm for polymer models gives improved sampling algorithms for spin systems at low temperatures on expander graphs and unbalanced bipartite graphs, among other applications.

Antonio Blanca, Sarah Cannon, and Will Perkins. Fast and Perfect Sampling of Subgraphs and Polymer Systems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 4:1-4:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{blanca_et_al:LIPIcs.APPROX/RANDOM.2022.4, author = {Blanca, Antonio and Cannon, Sarah and Perkins, Will}, title = {{Fast and Perfect Sampling of Subgraphs and Polymer Systems}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {4:1--4:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-249-5}, ISSN = {1868-8969}, year = {2022}, volume = {245}, editor = {Chakrabarti, Amit and Swamy, Chaitanya}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.4}, URN = {urn:nbn:de:0030-drops-171261}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.4}, annote = {Keywords: Random Sampling, perfect sampling, graphlets, polymer models, spin systems, percolation} }

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RANDOM

**Published in:** LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

We consider the problem of sampling from the ferromagnetic Potts and random-cluster models on a general family of random graphs via the Glauber dynamics for the random-cluster model. The random-cluster model is parametrized by an edge probability p ∈ (0,1) and a cluster weight q > 0. We establish that for every q ≥ 1, the random-cluster Glauber dynamics mixes in optimal Θ(nlog n) steps on n-vertex random graphs having a prescribed degree sequence with bounded average branching γ throughout the full high-temperature uniqueness regime p < p_u(q,γ).
The family of random graph models we consider includes the Erdős-Rényi random graph G(n,γ/n), and so we provide the first polynomial-time sampling algorithm for the ferromagnetic Potts model on Erdős-Rényi random graphs for the full tree uniqueness regime. We accompany our results with mixing time lower bounds (exponential in the largest degree) for the Potts Glauber dynamics, in the same settings where our Θ(n log n) bounds for the random-cluster Glauber dynamics apply. This reveals a novel and significant computational advantage of random-cluster based algorithms for sampling from the Potts model at high temperatures.

Antonio Blanca and Reza Gheissari. Sampling from Potts on Random Graphs of Unbounded Degree via Random-Cluster Dynamics. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 24:1-24:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{blanca_et_al:LIPIcs.APPROX/RANDOM.2022.24, author = {Blanca, Antonio and Gheissari, Reza}, title = {{Sampling from Potts on Random Graphs of Unbounded Degree via Random-Cluster Dynamics}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {24:1--24:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-249-5}, ISSN = {1868-8969}, year = {2022}, volume = {245}, editor = {Chakrabarti, Amit and Swamy, Chaitanya}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.24}, URN = {urn:nbn:de:0030-drops-171463}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.24}, annote = {Keywords: Potts model, random-cluster model, random graphs, Markov chains, mixing time, tree uniqueness} }

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RANDOM

**Published in:** LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

The Swendsen-Wang algorithm is a sophisticated, widely-used Markov chain for sampling from the Gibbs distribution for the ferromagnetic Ising and Potts models. This chain has proved difficult to analyze, due in part to the global nature of its updates. We present optimal bounds on the convergence rate of the Swendsen-Wang algorithm for the complete d-ary tree. Our bounds extend to the non-uniqueness region and apply to all boundary conditions. We show that the spatial mixing conditions known as Variance Mixing and Entropy Mixing, introduced in the study of local Markov chains by Martinelli et al. (2003), imply Ω(1) spectral gap and O(log n) mixing time, respectively, for the Swendsen-Wang dynamics on the d-ary tree. We also show that these bounds are asymptotically optimal. As a consequence, we establish Θ(log n) mixing for the Swendsen-Wang dynamics for all boundary conditions throughout the tree uniqueness region; in fact, our bounds hold beyond the uniqueness threshold for the Ising model, and for the q-state Potts model when q is small with respect to d. Our proofs feature a novel spectral view of the Variance Mixing condition inspired by several recent rapid mixing results on high-dimensional expanders and utilize recent work on block factorization of entropy under spatial mixing conditions.

Antonio Blanca, Zongchen Chen, Daniel Štefankovič, and Eric Vigoda. The Swendsen-Wang Dynamics on Trees. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 43:1-43:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{blanca_et_al:LIPIcs.APPROX/RANDOM.2021.43, author = {Blanca, Antonio and Chen, Zongchen and \v{S}tefankovi\v{c}, Daniel and Vigoda, Eric}, title = {{The Swendsen-Wang Dynamics on Trees}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)}, pages = {43:1--43:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-207-5}, ISSN = {1868-8969}, year = {2021}, volume = {207}, editor = {Wootters, Mary and Sanit\`{a}, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.43}, URN = {urn:nbn:de:0030-drops-147366}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.43}, annote = {Keywords: Markov Chains, mixing times, Ising and Potts models, Swendsen-Wang dynamics, trees} }

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RANDOM

**Published in:** LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

The random-cluster model is a unifying framework for studying random graphs, spin systems and electrical networks that plays a fundamental role in designing efficient Markov Chain Monte Carlo (MCMC) sampling algorithms for the classical ferromagnetic Ising and Potts models. In this paper, we study a natural non-local Markov chain known as the Chayes-Machta dynamics for the mean-field case of the random-cluster model, where the underlying graph is the complete graph on n vertices. The random-cluster model is parametrized by an edge probability p and a cluster weight q. Our focus is on the critical regime: p = p_c(q) and q ∈ (1,2), where p_c(q) is the threshold corresponding to the order-disorder phase transition of the model. We show that the mixing time of the Chayes-Machta dynamics is O(log n ⋅ log log n) in this parameter regime, which reveals that the dynamics does not undergo an exponential slowdown at criticality, a surprising fact that had been predicted (but not proved) by statistical physicists. This also provides a nearly optimal bound (up to the log log n factor) for the mixing time of the mean-field Chayes-Machta dynamics in the only regime of parameters where no non-trivial bound was previously known. Our proof consists of a multi-phased coupling argument that combines several key ingredients, including a new local limit theorem, a precise bound on the maximum of symmetric random walks with varying step sizes, and tailored estimates for critical random graphs. In addition, we derive an improved comparison inequality between the mixing time of the Chayes-Machta dynamics and that of the local Glauber dynamics on general graphs; this results in better mixing time bounds for the local dynamics in the mean-field setting.

Antonio Blanca, Alistair Sinclair, and Xusheng Zhang. The Critical Mean-Field Chayes-Machta Dynamics. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 47:1-47:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{blanca_et_al:LIPIcs.APPROX/RANDOM.2021.47, author = {Blanca, Antonio and Sinclair, Alistair and Zhang, Xusheng}, title = {{The Critical Mean-Field Chayes-Machta Dynamics}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)}, pages = {47:1--47:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-207-5}, ISSN = {1868-8969}, year = {2021}, volume = {207}, editor = {Wootters, Mary and Sanit\`{a}, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.47}, URN = {urn:nbn:de:0030-drops-147408}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.47}, annote = {Keywords: Markov Chains, Mixing Times, Random-cluster Model, Ising and Potts Models, Mean-field, Chayes-Machta Dynamics, Random Graphs} }

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RANDOM

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

The random-cluster (FK) model is a key tool for the study of phase transitions and for the design of efficient Markov chain Monte Carlo (MCMC) sampling algorithms for the Ising/Potts model. It is well-known that in the high-temperature region beta<beta_c(q) of the q-state Ising/Potts model on an n x n box Lambda_n of the integer lattice Z^2, spin correlations decay exponentially fast; this property holds even arbitrarily close to the boundary of Lambda_n and uniformly over all boundary conditions. A direct consequence of this property is that the corresponding single-site update Markov chain, known as the Glauber dynamics, mixes in optimal O(n^2 log{n}) steps on Lambda_{n} for all choices of boundary conditions. We study the effect of boundary conditions on the FK-dynamics, the analogous Glauber dynamics for the random-cluster model.
On Lambda_n the random-cluster model with parameters (p,q) has a sharp phase transition at p = p_c(q). Unlike the Ising/Potts model, the random-cluster model has non-local interactions which can be forced by boundary conditions: external wirings of boundary vertices of Lambda_n. We consider the broad and natural class of boundary conditions that are realizable as a configuration on Z^2 \ Lambda_n. Such boundary conditions can have many macroscopic wirings and impose long-range correlations even at very high temperatures (p << p_c(q)). In this paper, we prove that when q>1 and p != p_c(q) the mixing time of the FK-dynamics is polynomial in n for every realizable boundary condition. Previously, for boundary conditions that do not carry long-range information (namely wired and free), Blanca and Sinclair (2017) had proved that the FK-dynamics in the same setting mixes in optimal O(n^2 log n) time. To illustrate the difficulties introduced by general boundary conditions, we also construct a class of non-realizable boundary conditions that induce slow (stretched-exponential) convergence at high temperatures.

Antonio Blanca, Reza Gheissari, and Eric Vigoda. Random-Cluster Dynamics in Z^2: Rapid Mixing with General Boundary Conditions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 67:1-67:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{blanca_et_al:LIPIcs.APPROX-RANDOM.2019.67, author = {Blanca, Antonio and Gheissari, Reza and Vigoda, Eric}, title = {{Random-Cluster Dynamics in Z^2: Rapid Mixing with General Boundary Conditions}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {67:1--67:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.67}, URN = {urn:nbn:de:0030-drops-112827}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.67}, annote = {Keywords: Markov chain, mixing time, random-cluster model, Glauber dynamics, spatial mixing} }

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**Published in:** LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

The Swendsen-Wang dynamics is a popular algorithm for sampling from the Gibbs distribution for the ferromagnetic Ising model on a graph G=(V,E). The dynamics is a "global" Markov chain which is conjectured to converge to equilibrium in O(|V|^{1/4}) steps for any graph G at any (inverse) temperature beta. It was recently proved by Guo and Jerrum (2017) that the Swendsen-Wang dynamics has polynomial mixing time on any graph at all temperatures, yet there are few results providing o(|V|) upper bounds on its convergence time.
We prove fast convergence of the Swendsen-Wang dynamics on general graphs in the tree uniqueness region of the ferromagnetic Ising model. In particular, when beta < beta_c(d) where beta_c(d) denotes the uniqueness/non-uniqueness threshold on infinite d-regular trees, we prove that the relaxation time (i.e., the inverse spectral gap) of the Swendsen-Wang dynamics is Theta(1) on any graph of maximum degree d >= 3. Our proof utilizes a version of the Swendsen-Wang dynamics which only updates isolated vertices. We establish that this variant of the Swendsen-Wang dynamics has mixing time O(log{|V|}) and relaxation time Theta(1) on any graph of maximum degree d for all beta < beta_c(d). We believe that this Markov chain may be of independent interest, as it is a monotone Swendsen-Wang type chain. As part of our proofs, we provide modest extensions of the technology of Mossel and Sly (2013) for analyzing mixing times and of the censoring result of Peres and Winkler (2013). Both of these results are for the Glauber dynamics, and we extend them here to general monotone Markov chains. This class of dynamics includes for example the heat-bath block dynamics, for which we obtain new tight mixing time bounds.

Antonio Blanca, Zongchen Chen, and Eric Vigoda. Swendsen-Wang Dynamics for General Graphs in the Tree Uniqueness Region. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 32:1-32:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{blanca_et_al:LIPIcs.APPROX-RANDOM.2018.32, author = {Blanca, Antonio and Chen, Zongchen and Vigoda, Eric}, title = {{Swendsen-Wang Dynamics for General Graphs in the Tree Uniqueness Region}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)}, pages = {32:1--32:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-085-9}, ISSN = {1868-8969}, year = {2018}, volume = {116}, editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.32}, URN = {urn:nbn:de:0030-drops-94365}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.32}, annote = {Keywords: Swendsen-Wang dynamics, mixing time, relaxation time, spatial mixing, censoring} }

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**Published in:** LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

We consider the problem of sampling from the Potts model on random regular graphs. It is conjectured that sampling is possible when the temperature of the model is in the so-called uniqueness regime of the regular tree, but positive algorithmic results have been for the most part elusive. In this paper, for all integers q >= 3 and Delta >= 3, we develop algorithms that produce samples within error o(1) from the q-state Potts model on random Delta-regular graphs, whenever the temperature is in uniqueness, for both the ferromagnetic and antiferromagnetic cases.
The algorithm for the antiferromagnetic Potts model is based on iteratively adding the edges of the graph and resampling a bichromatic class that contains the endpoints of the newly added edge. Key to the algorithm is how to perform the resampling step efficiently since bichromatic classes can potentially induce linear-sized components. To this end, we exploit the tree uniqueness to show that the average growth of bichromatic components is typically small, which allows us to use correlation decay algorithms for the resampling step. While the precise uniqueness threshold on the tree is not known for general values of q and Delta in the antiferromagnetic case, our algorithm works throughout uniqueness regardless of its value.
In the case of the ferromagnetic Potts model, we are able to simplify the algorithm significantly by utilising the random-cluster representation of the model. In particular, we demonstrate that a percolation-type algorithm succeeds in sampling from the random-cluster model with parameters p,q on random Delta-regular graphs for all values of q >= 1 and p<p_c(q,Delta), where p_c(q,Delta) corresponds to a uniqueness threshold for the model on the Delta-regular tree. When restricted to integer values of q, this yields a simplified algorithm for the ferromagnetic Potts model on random Delta-regular graphs.

Antonio Blanca, Andreas Galanis, Leslie Ann Goldberg, Daniel Stefankovic, Eric Vigoda, and Kuan Yang. Sampling in Uniqueness from the Potts and Random-Cluster Models on Random Regular Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 33:1-33:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{blanca_et_al:LIPIcs.APPROX-RANDOM.2018.33, author = {Blanca, Antonio and Galanis, Andreas and Goldberg, Leslie Ann and Stefankovic, Daniel and Vigoda, Eric and Yang, Kuan}, title = {{Sampling in Uniqueness from the Potts and Random-Cluster Models on Random Regular Graphs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)}, pages = {33:1--33:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-085-9}, ISSN = {1868-8969}, year = {2018}, volume = {116}, editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.33}, URN = {urn:nbn:de:0030-drops-94371}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.33}, annote = {Keywords: sampling, Potts model, random regular graphs, phase transitions} }

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**Published in:** LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

The random-cluster model has been widely studied as a unifying framework for random graphs, spin systems and random spanning trees, but its dynamics have so far largely resisted analysis. In this paper we study a natural non-local Markov chain known as the Chayes-Machta dynamics for the mean-field case of the random-cluster model, and identify a critical regime (lambda_s,lambda_S) of the model parameter lambda in which the dynamics undergoes an exponential slowdown. Namely, we prove that the mixing time is Theta(log n) if lambda is not in [lambda_s,lambda_S], and e^Omega(sqrt{n}) when lambda is in (lambda_s,lambda_S). These results hold for all values of the second model parameter q > 1. In addition, we prove that the local heat-bath dynamics undergoes a similar exponential slowdown in (lambda_s,lambda_S).

Antonio Blanca and Alistair Sinclair. Dynamics for the Mean-field Random-cluster Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 528-543, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

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@InProceedings{blanca_et_al:LIPIcs.APPROX-RANDOM.2015.528, author = {Blanca, Antonio and Sinclair, Alistair}, title = {{Dynamics for the Mean-field Random-cluster Model}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {528--543}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.528}, URN = {urn:nbn:de:0030-drops-53227}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.528}, annote = {Keywords: random-cluster model, random graphs, Markov chains, statistical physics, dynamics} }

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