an:07003145
Zbl 07003145
Duminil-Copin, Hugo; Raoufi, Aran; Tassion, Vincent
Sharp phase transition for the random-cluster and Potts models via decision trees
EN
Ann. Math. (2) 189, No. 1, 75-99 (2019).
00426228
2019
j
60K35
percolation model; Potts model; randomized algorithm; sharp threshold; exponential decay
Summary: 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{\parindent=0.7cm\begin{itemize}\item[{\(\bullet\)}] For the Potts model on transitive graphs, correlations decay exponentially fast for \(\beta<\beta_c\).\item[{\(\bullet\)}] For the random-cluster model with cluster weight \(q\geq 1\) 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.\item[{\(\bullet\)}] For the random-cluster models with cluster weight \(q\geq 1\) 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.)
\end{itemize}} These results have many applications for the understanding of the subcritical (respectively disordered) phase of all these models. The techniques developed in this paper have potential to be extended to a wide class of models including the Ashkin-Teller model, continuum percolation models such as Voronoi percolation and Boolean percolation, super-level sets of massive Gaussian free field, and the random-cluster and Potts models with infinite range interactions.