On the Gibbs states of the non-critical Potts model on Z 2 Joint - PowerPoint PPT Presentation

On the Gibbs states of the non-critical Potts model on Z 2 Joint work with H. Duminil-Copin, D. Ioffe, and Y. Velenik. Loren Coquille Hausdorff Center für Mathematik, Universität Bonn July 10, 2014 Loren Coquille (HCM-Bonn) MAC2 Workshop – Paris July 10, 2014 1 / 34

Outline of the talk Finite volume measures 1 Phase transition 2 3 Infinite volume measures Known results 4 New result 5 Main ideas of the proof 6 Loren Coquille (HCM-Bonn) MAC2 Workshop – Paris July 10, 2014 2 / 34

Finite volume measures Outline of the talk Finite volume measures 1 Phase transition 2 3 Infinite volume measures Known results 4 New result 5 Main ideas of the proof 6 Loren Coquille (HCM-Bonn) MAC2 Workshop – Paris July 10, 2014 3 / 34

Finite volume measures Ising model Spin = state of a site x ∈ Z d σ x ∈ {− 1 , + 1 } Λ Hamiltonian = energy of a spin configuration � •• or •• contribution = -1 � H ( σ ) = − σ x σ y → •• or •• contribution = +1 x ∼ y Gibbs measure on {− 1 , + 1 } Λ : P Λ , T ( σ ) = 1 � − 1 � Z exp T H ( σ ) , T = temperature Loren Coquille (HCM-Bonn) MAC2 Workshop – Paris July 10, 2014 4 / 34

Finite volume measures Ising model / Potts model Spin = state of a site x ∈ Z d σ x ∈ {− 1 , + 1 } σ x ∈ { 1 , 2 . . . , q } Λ Hamiltonian = energy of a spin configuration � •• or •• contribution = -1 � H ( σ ) = − σ x σ y → •• or •• contribution = +1 x ∼ y � •• or •• or •• or . . . contribution = -1 � H ( σ ) = − δ σ x = σ y → •• or •• or •• or . . . contribution = 0 x ∼ y P Λ , T ( σ ) = 1 � − 1 � Z exp T H ( σ ) , T = temperature Loren Coquille (HCM-Bonn) MAC2 Workshop – Paris July 10, 2014 4 / 34

Finite volume measures Boundary conditions � � Ising : H ω ( σ ) = − σ x σ y − σ x ω y x ∼ y x ∈ Λ , y ∈ ∂ Λ x ∼ y � � Potts : H ω ( σ ) = − δ σ x = σ y − δ σ x = ω y x ∼ y x ∈ Λ , y ∈ ∂ Λ x ∼ y Loren Coquille (HCM-Bonn) MAC2 Workshop – Paris July 10, 2014 5 / 34

Finite volume measures Boundary conditions � � Ising : H ω ( σ ) = − σ x σ y − σ x ω y x ∼ y x ∈ Λ , y ∈ ∂ Λ x ∼ y � � Potts : H ω ( σ ) = − P ω δ σ x = σ y − δ σ x = ω y � Λ , T x ∼ y x ∈ Λ , y ∈ ∂ Λ x ∼ y Loren Coquille (HCM-Bonn) MAC2 Workshop – Paris July 10, 2014 5 / 34

Phase transition Outline of the talk Finite volume measures 1 Phase transition 2 3 Infinite volume measures Known results 4 New result 5 Main ideas of the proof 6 Loren Coquille (HCM-Bonn) MAC2 Workshop – Paris July 10, 2014 6 / 34

Phase transition Monochromatic boundary conditions Ising : P + Λ , T and P − Λ , T Potts : P i Λ , T , with i = 1 , . . . , q . T ≫ 1 T ≃ 0 Loren Coquille (HCM-Bonn) MAC2 Workshop – Paris July 10, 2014 7 / 34

Phase transition Phase transition Ising Potts There exists T c ( d , q ) ∈ ( 0 , ∞ ) s.t. There exists T c ( d ) ∈ ( 0 , ∞ ) s.t. If T > T c then If T > T c then Λ ↑ Z d E + Λ ↑ Z d P i lim Λ , T ( σ 0 ) = 0 lim Λ , T ( σ 0 = i ) = 1 / q If T < T c then If T < T c then Λ ↑ Z d E + Λ ↑ Z d P i lim Λ , T ( σ 0 ) > 0 lim Λ , T ( σ 0 = i ) > 1 / q Loren Coquille (HCM-Bonn) MAC2 Workshop – Paris July 10, 2014 8 / 34

Phase transition Phase transition Ising Potts There exists T c ( d ) ∈ ( 0 , ∞ ) s.t. There exists T c ( d , q ) ∈ ( 0 , ∞ ) s.t. If T > T c then If T > T c then E + P i T ( σ 0 ) = 0 T ( σ 0 = i ) = 1 / q If T < T c then If T < T c then − E − T = E + P i T ( σ 0 ) > 0 T ( σ 0 = i ) > 1 / q Remarks Existence of the monochromatic phases : FKG inequality for Ising, coupling with the random-clusters model for Potts. They are translation invariant. Loren Coquille (HCM-Bonn) MAC2 Workshop – Paris July 10, 2014 8 / 34

Phase transition Phase transition Ising Potts There exists T c ( d ) ∈ ( 0 , ∞ ) s.t. There exists T c ( d , q ) ∈ ( 0 , ∞ ) s.t. If T > T c then If T > T c then E + P i T ( σ 0 ) = 0 T ( σ 0 = i ) = 1 / q If T < T c then If T < T c then − E − T = E + P i T ( σ 0 ) > 0 T ( σ 0 = i ) > 1 / q Remarks Existence of the monochromatic phases : FKG inequality for Ising, coupling with the random-clusters model for Potts. They are translation invariant. Non-triviality of T c : Peierls in d = q = 2, monotonicity in d and in q . Loren Coquille (HCM-Bonn) MAC2 Workshop – Paris July 10, 2014 8 / 34

Infinite volume measures Outline of the talk Finite volume measures 1 Phase transition 2 3 Infinite volume measures Known results 4 New result 5 Main ideas of the proof 6 Loren Coquille (HCM-Bonn) MAC2 Workshop – Paris July 10, 2014 9 / 34

Infinite volume measures Infinite volume Gibbs measures Weak limits approach � accumulation points of sequences ( P ω n Λ n ) n � G T = with Λ n ↑ Z d as n → ∞ Weak topology : P ω n E ω n Λ n → P ⇔ Λ n ( f ) → E ( f ) ∀ f local Loren Coquille (HCM-Bonn) MAC2 Workshop – Paris July 10, 2014 10 / 34

Infinite volume measures Infinite volume Gibbs measures Weak limits approach � accumulation points of sequences ( P ω n Λ n ) n � G T = with Λ n ↑ Z d as n → ∞ Weak topology : P ω n E ω n Λ n → P ⇔ Λ n ( f ) → E ( f ) ∀ f local Dobrushin-Lanford-Ruelle approach � P : for all Λ ⋐ Z d , and P -a.e. ω, � ˜ G T = P ( σ | σ = ω on Λ c ) = P ω Λ ( σ ) Loren Coquille (HCM-Bonn) MAC2 Workshop – Paris July 10, 2014 10 / 34

Infinite volume measures Infinite volume Gibbs measures Weak limits approach � accumulation points of sequences ( P ω n Λ n ) n � G T = with Λ n ↑ Z d as n → ∞ Weak topology : P ω n E ω n Λ n → P ⇔ Λ n ( f ) → E ( f ) ∀ f local Dobrushin-Lanford-Ruelle approach � P : for all Λ ⋐ Z d , and P -a.e. ω, � ˜ G T = P ( σ | σ = ω on Λ c ) = P ω Λ ( σ ) ˜ G T is a simplex, whose extremal elements belong to G T ! Loren Coquille (HCM-Bonn) MAC2 Workshop – Paris July 10, 2014 10 / 34

Known results Outline of the talk Finite volume measures 1 Phase transition 2 3 Infinite volume measures Known results 4 New result 5 Main ideas of the proof 6 Loren Coquille (HCM-Bonn) MAC2 Workshop – Paris July 10, 2014 11 / 34

Known results Known results for the Ising model 1972 For 0 < T ≪ T c and d = 2 resp. 3, [Galavotti] [Dobrushin] P ± = ( P + + P − ) / 2 , P ± is not translation invariant + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + O ( √ n ) + + + O (1) + + + + − + + + − − − − − − − − − − − − − − − − − − − − − −− − − − − − − − − − − − − − − − − − − − − − n n Loren Coquille (HCM-Bonn) MAC2 Workshop – Paris July 10, 2014 12 / 34

Known results Known results for the Ising model 1980-81 In d = 2 for all T > 0 [Aizenman] [Higuchi] α P + T + ( 1 − α ) P − � � G T = T , with α ∈ [ 0 , 1 ] T ≥ T c ⇒ G T = { P T } • G T = [ P − T , P + T < T c ⇒ T ] (Proof by contradiction, in the infinite volume setting, gives little information about large but finite volumes.) Loren Coquille (HCM-Bonn) MAC2 Workshop – Paris July 10, 2014 13 / 34

Known results Known results for the Ising model 1979 In d = 2 and for T ≪ T c [Higuchi] 2005 In d = 2 and T < T c [Greenberg, Ioffe] Under diffusive scaling, the Dobrushin interface weakly converges to a Brownian bridge. Loren Coquille (HCM-Bonn) MAC2 Workshop – Paris July 10, 2014 14 / 34

Known results Known results for the Potts model 1986 For d ≥ 2, q > q 0 ( d ) , and T < T c [Martirosian] � q i = 1 α i P i If P ∈ G T is translation invariant, then P = T . Loren Coquille (HCM-Bonn) MAC2 Workshop – Paris July 10, 2014 15 / 34

Known results Known results for the Potts model 1986 For d ≥ 2, q > q 0 ( d ) , and T < T c [Martirosian] � q i = 1 α i P i If P ∈ G T is translation invariant, then P = T . 2008 For d = 2, and T < T c (*) [Campanino, Ioffe, Velenik] Under diffusive scaling, the Dobrushin interface weakly converges to a Brownian bridge. Loren Coquille (HCM-Bonn) MAC2 Workshop – Paris July 10, 2014 15 / 34

New result Outline of the talk Finite volume measures 1 Phase transition 2 3 Infinite volume measures Known results 4 New result 5 Main ideas of the proof 6 Loren Coquille (HCM-Bonn) MAC2 Workshop – Paris July 10, 2014 16 / 34

New result Characterization of G T for the Potts model on Z 2 Theorem (C., Duminil-Copin, Ioffe, Velenik) For q ≥ 2 and T < T c ( q ) , �� q � q � i = 1 α i P i G T = T , with i = 1 α i = 1 Loren Coquille (HCM-Bonn) MAC2 Workshop – Paris July 10, 2014 17 / 34