# zbMATH — the first resource for mathematics

Sharp phase transition for the random-cluster and Potts models via decision trees. (English) Zbl 07003145
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
$$\bullet$$
For the Potts model on transitive graphs, correlations decay exponentially fast for $$\beta<\beta_c$$.
$$\bullet$$
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.
$$\bullet$$
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.)
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.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory
Full Text:
##### References:
 [1] Aizenman, Michael; Barsky, David J., Sharpness of the phase transition in percolation models, Comm. Math. Phys.. Communications in Mathematical Physics, 108, 489-526, (1987) · Zbl 0618.60098 [2] Aizenman, M.; Barsky, D. J.; Fern\'andez, R., The phase transition in a general class of Ising-type models is sharp, J. Statist. Phys.. Journal of Statistical Physics, 47, 343-374, (1987) [3] Aizenman, Michael; Duminil-Copin, Hugo; Sidoravicius, Vladas, Random currents and continuity of Ising model’s spontaneous magnetization, Comm. Math. Phys.. Communications in Mathematical Physics, 334, 719-742, (2015) · Zbl 1315.82004 [4] Aizenman, M.; Fern\'andez, R., On the critical behavior of the magnetization in high-dimensional Ising models, J. Statist. Phys.. Journal of Statistical Physics, 44, 393-454, (1986) · Zbl 0629.60106 [5] Alexander, Kenneth S., Mixing properties and exponential decay for lattice systems in finite volumes, Ann. Probab.. The Annals of Probability, 32, 441-487, (2004) · Zbl 1048.60080 [6] Baxter, Rodney J., Exactly Solved Models in Statistical Mechanics, xii+486 pp., (1982) · Zbl 1201.60091 [7] Beffara, Vincent; Duminil-Copin, Hugo, The self-dual point of the two-dimensional random-cluster model is critical for $$q\geq 1$$, Probab. Theory Related Fields. Probability Theory and Related Fields, 153, 511-542, (2012) · Zbl 1257.82014 [8] Biskup, Marek; Chayes, Lincoln, Rigorous analysis of discontinuous phase transitions via mean-field bounds, Comm. Math. Phys.. Communications in Mathematical Physics, 238, 53-93, (2003) · Zbl 1051.82008 [9] Bollob\'as, B\'ela; Riordan, Oliver, Percolation on self-dual polygon configurations. An Irregular Mind, Bolyai Soc. Math. Stud., 21, 131-217, (2010) · Zbl 1215.05021 [10] Bourgain, Jean; Kahn, Jeff; Kalai, Gil; Katznelson, Yitzhak; Linial, Nathan, The influence of variables in product spaces, Israel J. Math.. Israel Journal of Mathematics, 77, 55-64, (1992) · Zbl 0771.60002 [11] Buhrman, Harry; de Wolf, Ronald, Complexity measures and decision tree complexity: a survey, Theoret. Comput. Sci.. Theoretical Computer Science, 288, 21-43, (2002) · Zbl 1061.68058 [12] Campanino, Massimo; Ioffe, Dmitry; Velenik, Yvan, Fluctuation theory of connectivities for subcritical random cluster models, Ann. Probab.. The Annals of Probability, 36, 1287-1321, (2008) · Zbl 1160.60026 [13] Duminil-Copin, Hugo, Parafermionic Observables and their Applications to Planar Statistical Physics Models, Ensaios Matem\'aticos [Mathematical Surveys], 25, ii+371 pp., (2013) · Zbl 1298.82001 [14] Duminil-Copin, Hugo; Manolescu, Ioan, The phase transitions of the planar random-cluster and Potts models with $$q\ge1$$ are sharp, Probab. Theory Related Fields. Probability Theory and Related Fields, 164, 865-892, (2016) · Zbl 1356.60167 [15] Duminil-Copin, H.; Gagnebin, M.; Harel, M.; Manolescu, I.; Tassion, V., Discontinuity of the phase transition for the planar random-cluster and Potts models with $$q> 4$$ [16] Duminil-Copin, Hugo; Raoufi, Aran; Tassion, Vincent, A new computation of the critical point for the planar random-cluster model with $$q\ge1$$, Ann. Inst. Henri Poincar\'e Probab. Stat.. Annales de l’Institut Henri Poincar\'e Probabilit\'es et Statistiques, 54, 422-436, (2018) · Zbl 1395.82043 [17] Duminil-Copin, Hugo; Raoufi, Aran; Tassion, Vincent, Exponential decay of connection probabilities for subcritical Voronoi percolation in $$\mathbb{R}^d$$, (2017) · Zbl 1395.82043 [18] Duminil-Copin, Hugo; Raoufi, Aran; Tassion, Vincent, Subcritical phase of $$d$$-dimensional Poisson-Boolean percolation and its vacant set, (2018) · Zbl 1395.82043 [19] Duminil-Copin, Hugo; Sidoravicius, Vladas; Tassion, Vincent, Continuity of the phase transition for planar random-cluster and Potts models with $$1 \leq q \leq 4$$, Comm. Math. Phys.. Communications in Mathematical Physics, 349, 47-107, (2017) · Zbl 1357.82011 [20] Duminil-Copin, Hugo; Tassion, Vincent, A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model, Comm. Math. Phys.. Communications in Mathematical Physics, 343, 725-745, (2016) · Zbl 1342.82026 [21] Fortuin, C. M.; Kasteleyn, P. W., On the random-cluster model. I. Introduction and relation to other models, Physica, 57, 536-564, (1972) [22] Garban, Christophe; Steif, Jeffrey E., Noise Sensitivity of Boolean Functions and Percolation, Inst. Math. Stat. Textbooks, 5, xvii+203 pp., (2015) · Zbl 1355.06001 [23] Graham, B. T.; Grimmett, G. R., Influence and sharp-threshold theorems for monotonic measures, Ann. Probab.. The Annals of Probability, 34, 1726-1745, (2006) · Zbl 1115.60099 [24] Grimmett, Geoffrey, The Random-Cluster Model, Grundlehren Math. Wiss., 333, xiv+377 pp., (2006) · Zbl 1122.60087 [25] Kahn, J.; Kalai, G.; Linial, N., The influence of variables on Boolean functions [26] Koteck\'y, R.; Shlosman, S. B., First-order phase transitions in large entropy lattice models, Comm. Math. Phys.. Communications in Mathematical Physics, 83, 493-515, (1982) [27] Laanait, Lahoussine; Messager, Alain; Miracle-Sol\'e, Salvador; Ruiz, Jean; Shlosman, Senya, Interfaces in the Potts model. I. Pirogov-Sinai theory of the Fortuin-Kasteleyn representation, Comm. Math. Phys.. Communications in Mathematical Physics, 140, 81-91, (1991) · Zbl 0734.60108 [28] Manolescu, Ioan; Raoufi, Aran, The phase transitions of the random-cluster and Potts models on slabs with $$q\geq 1$$ are sharp, Electron. J. Probab.. Electronic Journal of Probability, 23, 63-25, (2018) · Zbl 1410.60096 [29] Martinelli, Fabio, Lectures on Glauber dynamics for discrete spin models. Lectures on Probability Theory and Statistics, Lecture Notes in Math., 1717, 93-191, (1999) · Zbl 1051.82514 [30] Men\cprimeshikov, M. V., Coincidence of critical points in percolation problems, Dokl. Akad. Nauk SSSR. Doklady Akademii Nauk SSSR, 288, 1308-1311, (1986) [31] O’Donnell, R.; Saks, M.; Schramm, O.; Servedio, R., Every decision tree has an influential variable, (2005) [32] O’Donnell, Ryan, Analysis of Boolean Functions, xx+423 pp., (2014) · Zbl 1336.94096 [33] Potts, R. B., Some generalized order-disorder transformations, Proc. Cambridge Philos. Soc., 48, 106-109, (1952) · Zbl 0048.45601 [34] Schramm, Oded; Steif, Jeffrey E., Quantitative noise sensitivity and exceptional times for percolation, Ann. of Math. (2). Annals of Mathematics. Second Series, 171, 619-672, (2010) · Zbl 1213.60160 [35] Sheffield, Scott, Random Surfaces, Ast\'erisque, vi+175 pp., (2005) · Zbl 1104.60002 [36] Yao, Andrew Chi Chih, Probabilistic computations: toward a unified measure of complexity (extended abstract). 18th Annual Symposium on Foundations of Computer Science (Providence, R.I., 1977), 222-227, (1977)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.