### Abstract

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 Δ ≥ 3, we develop algorithms that produce samples within error o(1) from the q-state Potts model on random Δ-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 Δ 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 Δ-regular graphs for all values of q 1 and p < pc(q, Δ), where pc(q, Δ) corresponds to a uniqueness threshold for the model on the Δ-regular tree. When restricted to integer values of q, this yields a simplified algorithm for the ferromagnetic Potts model on random Δ-regular graphs.

Original language | English (US) |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018 |

Editors | Eric Blais, Jose D. P. Rolim, David Steurer, Klaus Jansen |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Print) | 9783959770859 |

DOIs | |

State | Published - Aug 1 2018 |

Event | 21st International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2018 and the 22nd International Workshop on Randomization and Computation, RANDOM 2018 - Princeton, United States Duration: Aug 20 2018 → Aug 22 2018 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 116 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 21st International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2018 and the 22nd International Workshop on Randomization and Computation, RANDOM 2018 |
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Country | United States |

City | Princeton |

Period | 8/20/18 → 8/22/18 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Software

### Cite this

*Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018*[33] (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 116). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.33

}

*Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018.*, 33, Leibniz International Proceedings in Informatics, LIPIcs, vol. 116, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 21st International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2018 and the 22nd International Workshop on Randomization and Computation, RANDOM 2018, Princeton, United States, 8/20/18. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.33

**Sampling in uniqueness from the potts and random-cluster models on random regular graphs.** / Pimentel, Antonio Blanca; Galanis, Andreas; Goldberg, Leslie Ann; Štefankovic, Daniel; Vigoda, Eric; Yang, Kuan.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Sampling in uniqueness from the potts and random-cluster models on random regular graphs

AU - Pimentel, Antonio Blanca

AU - Galanis, Andreas

AU - Goldberg, Leslie Ann

AU - Štefankovic, Daniel

AU - Vigoda, Eric

AU - Yang, Kuan

PY - 2018/8/1

Y1 - 2018/8/1

N2 - 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 Δ ≥ 3, we develop algorithms that produce samples within error o(1) from the q-state Potts model on random Δ-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 Δ 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 Δ-regular graphs for all values of q 1 and p < pc(q, Δ), where pc(q, Δ) corresponds to a uniqueness threshold for the model on the Δ-regular tree. When restricted to integer values of q, this yields a simplified algorithm for the ferromagnetic Potts model on random Δ-regular graphs.

AB - 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 Δ ≥ 3, we develop algorithms that produce samples within error o(1) from the q-state Potts model on random Δ-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 Δ 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 Δ-regular graphs for all values of q 1 and p < pc(q, Δ), where pc(q, Δ) corresponds to a uniqueness threshold for the model on the Δ-regular tree. When restricted to integer values of q, this yields a simplified algorithm for the ferromagnetic Potts model on random Δ-regular graphs.

UR - http://www.scopus.com/inward/record.url?scp=85052442828&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85052442828&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.APPROX-RANDOM.2018.33

DO - 10.4230/LIPIcs.APPROX-RANDOM.2018.33

M3 - Conference contribution

AN - SCOPUS:85052442828

SN - 9783959770859

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018

A2 - Blais, Eric

A2 - Rolim, Jose D. P.

A2 - Steurer, David

A2 - Jansen, Klaus

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

ER -