First order phase transition for the Random Cluster model with q > - PowerPoint PPT Presentation

First order phase transition for the Random Cluster model with q > 4 Ioan Manolescu joint work with: Hugo Duminil-Copin, Maxime Gagnebin, Matan Harel, Vincent Tassion University of Fribourg 14th February 2017 Diablerets Ioan Manolescu (University of Fribourg) Random Cluster q > 4 14th Feb. 2017 1 / 13

Setting: G is a finite subgraph of Z 2 . Random Cluster model: parameters q ≥ 1 and p ∈ [0 , 1] on G = ( V , E ): 1 ω ∈ { 0 , 1 } E with probability p o ( ω ) (1 − p ) c ( ω ) q k ( ω ) . Φ p , G , q ( ω ) = Z p , G , q Ioan Manolescu (University of Fribourg) Random Cluster q > 4 14th Feb. 2017 2 / 13

Setting: G is a finite subgraph of Z 2 . Random Cluster model: parameters q ≥ 1 and p ∈ [0 , 1] on G = ( V , E ): 1 ω ∈ { 0 , 1 } E with probability Φ 0 / 1 p o ( ω ) (1 − p ) c ( ω ) q k 0 / 1 ( ω ) . p , G , q ( ω ) = Z p , G , q Ioan Manolescu (University of Fribourg) Random Cluster q > 4 14th Feb. 2017 2 / 13

Setting: G is a finite subgraph of Z 2 . Random Cluster model: parameters q ≥ 1 and p ∈ [0 , 1] on G = ( V , E ): 1 ω ∈ { 0 , 1 } E with probability Φ 0 / 1 p o ( ω ) (1 − p ) c ( ω ) q k 0 / 1 ( ω ) . p , G , q ( ω ) = Z p , G , q Infinite volume measures on Z 2 may be defined by taking limits: Φ 0 G → Z 2 Φ 0 Φ 1 G → Z 2 Φ 1 Φ 0 p , q ≤ Φ 1 p , G , q − − − − → and p , G , q − − − − → p , q . p , q . p , q Ioan Manolescu (University of Fribourg) Random Cluster q > 4 14th Feb. 2017 2 / 13

Setting: G is a finite subgraph of Z 2 . Random Cluster model: parameters q ≥ 1 and p ∈ [0 , 1] on G = ( V , E ): 1 ω ∈ { 0 , 1 } E with probability Φ 0 / 1 p o ( ω ) (1 − p ) c ( ω ) q k 0 / 1 ( ω ) . p , G , q ( ω ) = Z p , G , q Infinite volume measures on Z 2 may be defined by taking limits: Φ 0 G → Z 2 Φ 0 Φ 1 G → Z 2 Φ 1 Φ 0 p , q ≤ Φ 1 p , G , q − − − − → and p , G , q − − − − → p , q . p , q . p , q Phase transition in terms of infinite cluster (Φ p , q increasing in p ) ? subcritical phase supercritical phase 0 1 p c no infinite cluster, infinite cluster exists, connections decay exponentially finite clusters decay exponentially Ioan Manolescu (University of Fribourg) Random Cluster q > 4 14th Feb. 2017 2 / 13

Setting: G is a finite subgraph of Z 2 . Random Cluster model: parameters q ≥ 1 and p ∈ [0 , 1] on G = ( V , E ): 1 ω ∈ { 0 , 1 } E with probability Φ 0 / 1 p o ( ω ) (1 − p ) c ( ω ) q k 0 / 1 ( ω ) . p , G , q ( ω ) = Z p , G , q Infinite volume measures on Z 2 may be defined by taking limits: Φ 0 G → Z 2 Φ 0 Φ 1 G → Z 2 Φ 1 Φ 0 p , q ≤ Φ 1 p , G , q − − − − → and p , G , q − − − − → p , q . p , q . p , q Phase transition in terms of infinite cluster (Φ p , q increasing in p ) ? subcritical phase supercritical phase 0 1 p c no infinite cluster, infinite cluster exists, connections decay exponentially finite clusters decay exponentially Theorem (Beffara, Duminil-Copin 2012) √ q On Z 2 , p c = 1+ √ q (in other words p c = p sd , the self-dual parameter). Ioan Manolescu (University of Fribourg) Random Cluster q > 4 14th Feb. 2017 2 / 13

φ p (0 ↔ ∞ ) φ p (0 ↔ ∞ ) 1 1 [ ] 0 0 p c 1 1 p c Theorem (Duminil-Copin, Sidoravicius, Tassion 2015) Phase transition: either one or the other. Ioan Manolescu (University of Fribourg) Random Cluster q > 4 14th Feb. 2017 3 / 13

φ p (0 ↔ ∞ ) φ p (0 ↔ ∞ ) 1 1 Two critical measures φ 1 p c - supercritical [ φ p c - critical φ 0 p c - subcritical ] 0 0 p c 1 1 p c Theorem (Duminil-Copin, Sidoravicius, Tassion 2015) Phase transition: either one or the other. Continuous phase transition: . . . or discontinuous: φ 0 p c = φ 1 φ 0 p c � = φ 1 p c ; p c ; in φ p c connections decrease in φ 0 p c connections decrease polynomially; exponentially, no infinite cluster for φ p c ; infinite cluster in φ 1 p c . strong RSW type estimates. Ioan Manolescu (University of Fribourg) Random Cluster q > 4 14th Feb. 2017 3 / 13

φ p (0 ↔ ∞ ) φ p (0 ↔ ∞ ) 1 1 Two critical measures φ 1 p c - supercritical [ φ p c - critical φ 0 p c - subcritical ] 0 0 p c 1 1 p c Theorem (Duminil-Copin, Sidoravicius, Tassion 2015) Phase transition: either one or the other. Continuous phase transition: . . . or discontinuous: φ 0 p c = φ 1 φ 0 p c � = φ 1 p c ; p c ; in φ p c connections decrease in φ 0 p c connections decrease polynomially; exponentially, no infinite cluster for φ p c ; infinite cluster in φ 1 p c . strong RSW type estimates. When q ∈ [1 , 4] Ioan Manolescu (University of Fribourg) Random Cluster q > 4 14th Feb. 2017 3 / 13

φ p (0 ↔ ∞ ) φ p (0 ↔ ∞ ) 1 1 Two critical measures φ 1 p c - supercritical [ φ p c - critical φ 0 p c - subcritical ] 0 0 p c 1 1 p c Theorem (Duminil-Copin, Sidoravicius, Tassion 2015) Phase transition: either one or the other. Continuous phase transition: . . . or discontinuous: φ 0 p c = φ 1 φ 0 p c � = φ 1 p c ; p c ; in φ p c connections decrease in φ 0 p c connections decrease polynomially; exponentially, no infinite cluster for φ p c ; infinite cluster in φ 1 p c . strong RSW type estimates. (HDC,MH,MG,IM,VT 2017): When q ∈ [1 , 4] . . . when q > 4 Ioan Manolescu (University of Fribourg) Random Cluster q > 4 14th Feb. 2017 3 / 13

Theorem (H. Duminil-Copin, M. Gagnebin, M. Harel, I.M., V. Tassion) The phase transition of RCM on the square lattice with q > 4 is discontinuous . Moreover, if λ > 0 satisfies cosh( λ ) = √ q / 2 , then   ∂ Λ n ∞ n →∞ − 1 ξ ( q ) − 1 = lim ( − 1) k � n log φ 0    = λ + 2 tanh( k λ ) > 0 . p c , q   k  k =1 π 2 � � ξ ( q ) − 1 ∼ 8 exp As q ց 4 , − √ q − 4 .   ∂ Λ n n � � φ 0    = exp − ξ ( q ) + o ( n ) p c , q    Ioan Manolescu (University of Fribourg) Random Cluster q > 4 14th Feb. 2017 4 / 13

Relation the six vertex model. a a b b c c Ioan Manolescu (University of Fribourg) Random Cluster q > 4 14th Feb. 2017 5 / 13

A brief introduction to the six vertex model. Configurations: On a part of Z 2 : orient each edge s.t. each vertex has exactly two incoming edges . a a b b c c a n 1 + n 2 · b n 3 + n 4 · c n 5 + n 6 Weight: Ioan Manolescu (University of Fribourg) Random Cluster q > 4 14th Feb. 2017 6 / 13

A brief introduction to the six vertex model. Configurations: On a part of Z 2 : orient each edge s.t. each vertex has exactly two incoming edges . a a b b c c 1 a n 1 + n 2 · b n 3 + n 4 · c n 5 + n 6 Probability: Z 6 V Ioan Manolescu (University of Fribourg) Random Cluster q > 4 14th Feb. 2017 6 / 13

A brief introduction to the six vertex model. Configurations: On a part of Z 2 : orient each edge s.t. each vertex has exactly two incoming edges . a a b b c c 1 a n 1 + n 2 · b n 3 + n 4 · c n 5 + n 6 Probability: Z 6 V We limit ourselves to a = b = 1 and c ≥ 2 (∆= a 2 + b 2 − c 2 < − 1: anti-ferroelectric phase). 2 ab Ioan Manolescu (University of Fribourg) Random Cluster q > 4 14th Feb. 2017 6 / 13

A brief introduction to the six vertex model. Configurations: On a part of Z 2 : orient each edge s.t. each vertex has exactly two incoming edges . a a b b c c 1 c n 5 + n 6 Probability: Z 6 V We limit ourselves to a = b = 1 and c ≥ 2 (∆= a 2 + b 2 − c 2 < − 1: anti-ferroelectric phase). 2 ab M Torus T N , M , with M → ∞ then N → ∞ . N Ioan Manolescu (University of Fribourg) Random Cluster q > 4 14th Feb. 2017 6 / 13

A brief introduction to the six vertex model. Configurations: On a part of Z 2 : orient each edge s.t. each vertex has exactly two incoming edges . a a b b c c 1 c n 5 + n 6 Probability: Z 6 V We limit ourselves to a = b = 1 and c ≥ 2 (∆= a 2 + b 2 − c 2 < − 1: anti-ferroelectric phase). 2 ab M Torus T N , M , with M → ∞ then N → ∞ . N Ioan Manolescu (University of Fribourg) Random Cluster q > 4 14th Feb. 2017 6 / 13

A brief introduction to the six vertex model. Configurations: On a part of Z 2 : orient each edge s.t. each vertex has exactly two incoming edges . a a b b c c 1 c n 5 + n 6 Probability: Z 6 V We limit ourselves to a = b = 1 and c ≥ 2 (∆= a 2 + b 2 − c 2 < − 1: anti-ferroelectric phase). 2 ab M Torus T N , M , with M → ∞ then N → ∞ . N Ioan Manolescu (University of Fribourg) Random Cluster q > 4 14th Feb. 2017 6 / 13

From random cluster to six vertex. Ioan Manolescu (University of Fribourg) Random Cluster q > 4 14th Feb. 2017 7 / 13

From random cluster to six vertex. ω Ioan Manolescu (University of Fribourg) Random Cluster q > 4 14th Feb. 2017 7 / 13

From random cluster to six vertex. ω ( ℓ ) ω Ioan Manolescu (University of Fribourg) Random Cluster q > 4 14th Feb. 2017 7 / 13

From random cluster to six vertex. ω ( ℓ ) ω ω Ioan Manolescu (University of Fribourg) Random Cluster q > 4 14th Feb. 2017 7 / 13

From random cluster to six vertex. ω ( ℓ ) ω ω � ω Ioan Manolescu (University of Fribourg) Random Cluster q > 4 14th Feb. 2017 7 / 13

From random cluster to six vertex. ω ( ℓ ) ω ω � ω w RC ( ω ) = p o ( ω ) (1 − p sd ) c ( ω ) q k ( ω ) sd Ioan Manolescu (University of Fribourg) Random Cluster q > 4 14th Feb. 2017 7 / 13