TY - JOUR

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

AU - Blanca, Antonio

AU - Galanis, Andreas

AU - Goldberg, Leslie Ann

AU - Štefankovič, Daniel

AU - Vigoda, Eric

AU - Yang, Kuan

N1 - Funding Information:
\ast Received by the editors October 9, 2018; accepted for publication (in revised form) December 19, 2019; published electronically March 23, 2020. A preliminary short version of the manuscript (without the proofs) appeared in the proceedings of RANDOM/APPROX 2018. https://doi.org/10.1137/18M1219722 Funding: The first and fifth authors' research was supported in part by NSF grants CCF-1617306 and CCF-1563838. The fourth author's research was supported in part by NSF grant CCF-1563757. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013), ERC grant agreement 334828. The paper reflects only the authors' views and not the views of the ERC or the European Commission. The European Union is not liable for any use that may be made of the information contained therein.

PY - 2020

Y1 - 2020

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 socalled 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 utilizing 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 socalled 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 utilizing 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.

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U2 - 10.1137/18M1219722

DO - 10.1137/18M1219722

M3 - Article

AN - SCOPUS:85091400782

VL - 34

SP - 742

EP - 793

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 1

ER -