Abstract
We prove an inequality on decision trees on monotonic measures which generalizes the OSSS inequality on product spaces. As an application, we use this inequality to prove a number of new results on lattice spin models and their random-cluster representations. More precisely, we prove that
For the Potts model on transitive graphs, correlations decay exponentially fast for $\beta \lt \beta_c$.
For the random-cluster model with cluster weight $q\ge 1$ on transitive graphs, correlations decay exponentially fast in the subcritical regime and the cluster-density satisfies the mean-field lower bound in the supercritical regime.
For the random-cluster models with cluster weight $q\ge 1$ on planar quasi-transitive graphs $\mathbb{G}$, $$ \frac{p_c(\mathbb{G})p_c(\mathbb{G}^\ast)}{(1-p_c(\mathbb{G}))(1-p_c(\mathbb{G}^\ast))}=q. $$ As a special case, we obtain the value of the critical point for the square, triangular and hexagonal lattices. (This provides a short proof of a result of Beffara and the first author dating from 2012.)
Citation
Hugo Duminil-Copin. Aran Raoufi. Vincent Tassion. "Sharp phase transition for the random-cluster and Potts models via decision trees." Ann. of Math. (2) 189 (1) 75 - 99, January 2019. https://doi.org/10.4007/annals.2019.189.1.2
Information