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October 2020 Rarity of extremal edges in random surfaces and other theoretical applications of cluster algorithms
Omri Cohen-Alloro, Ron Peled
Ann. Appl. Probab. 30(5): 2439-2464 (October 2020). DOI: 10.1214/20-AAP1562


Motivated by questions on the delocalization of random surfaces, we prove that random surfaces satisfying a Lipschitz constraint rarely develop extremal gradients. Previous proofs of this fact relied on reflection positivity and were thus limited to random surfaces defined on highly symmetric graphs, whereas our argument applies to general graphs. Our proof makes use of a cluster algorithm and reflection transformation for random surfaces of the type introduced by Swendsen–Wang, Wolff and Evertz et al. We discuss the general framework for such cluster algorithms, reviewing several particular cases with emphasis on their use in obtaining theoretical results. Two additional applications are presented: A reflection principle for random surfaces and a proof that pair correlations in the spin $O(n)$ model have monotone densities, strengthening Griffiths’ first inequality for such correlations.


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Omri Cohen-Alloro. Ron Peled. "Rarity of extremal edges in random surfaces and other theoretical applications of cluster algorithms." Ann. Appl. Probab. 30 (5) 2439 - 2464, October 2020.


Received: 1 February 2018; Revised: 1 December 2019; Published: October 2020
First available in Project Euclid: 15 September 2020

MathSciNet: MR4149533
Digital Object Identifier: 10.1214/20-AAP1562

Primary: 82B05 , 82B20 , 82B41

Keywords: cluster algorithms , extremal gradients , Random surfaces , Swendsen–Wang algorithm , Wolff algorithm

Rights: Copyright © 2020 Institute of Mathematical Statistics


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Vol.30 • No. 5 • October 2020
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