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Motivated by questions on the delocalization of random surfaces, we provethat random surfaces satisfying a Lipschitz constraint rarely develop extremalgradients. Previous proofs of this fact relied on reflection positivity andwere thus limited to random surfaces defined on highly symmetric graphs,whereas our argument applies to general graphs. Our proof makes use of acluster algorithm and reflection transformation for random surfaces of the typeintroduced by Swendsen-Wang, Wolff and Evertz et al. We discuss the generalframework for such cluster algorithms, reviewing several particular cases withemphasis on their use in obtaining theoretical results. Two additionalapplications are presented: A reflection principle for random surfaces and aproof that pair correlations in the spin $O(n)$ model have monotone densities,strengthening Griffiths' first inequality for such correlations.

Rarity of extremal edges in random surfaces and other theoretical applications of cluster algorithms