One-sided versus two-sided measures: the lack of continuity Eric O. - PowerPoint PPT Presentation

One-sided versus two-sided measures: the lack of continuity Eric O. Endo 1 Department of Mathematics NYU Shanghai, China Jointly with: Rodrigo Bissacot (IME-USP, Brazil) Aernout C. D. van Enter (RUG, the Netherlands) Arnaud Le Ny (Université Paris-Est (UPEC), France) G2D2 - Yichang, China 1 e-mail address: eoe2@nyu.edu 1 / 14

g -function A : finite set (e.g. {− 1 , 1 } ); Set of sequences (Full shift) A Z − = { ( . . . , x − 1 , x 0 ) : x n ∈ A } ; T : A Z − → A Z − be the shift defined by T ( . . . , x − 1 , x 0 ) = ( . . . , x − 2 , x − 1 ) . g -functions: Continuous positive functions g : A Z − → ( 0 , 1 ) such that � for all x ∈ A Z − , g ( y ) = 1 . y ∈ T − 1 x 2 / 14

Ruelle Operator Ruelle operator: For a fixed g -function, � ( L g h )( x ) = g ( y ) h ( y ) , y ∈ T − 1 x for all continuous function h on A Z − . Dual of the Ruelle Operator: Defined on the set of probability measures by � � A Z − h ( x ) L ∗ g µ ( d x ) = A Z − ( L g h )( x ) µ ( d x ) for all continuous functions h on A Z − . 3 / 14

g -measure Definition A probability measure µ is a g -measure if there is a g -function g such that L ∗ g µ = µ. g -measure is translation-invariant. Can be extended to A Z . Ledrappier: µ is a g -measure if, and only if, for every ω 0 ∈ A , E µ [ ✶ σ 0 = ω 0 |F < 0 ] ( ω ) = g ( ω Z − ) µ − a . e . g -measure is also called Chain of Infinite Order. Markov chain given all past. 4 / 14

Gibbs measures Finite subset of Z : Λ ⋐ Z (Interval). Configuration: σ Λ ∈ A Λ . Boundary condition: ω Λ c ∈ A Λ c . F Λ be the sigma algebra on Λ generated by the cylinder sets. Interaction: Collection Φ = (Φ X ) X ⋐ Z of F X -measurable functions Φ X : A X → R over all X ⋐ Z . Hamiltonian: Sum of interactions H ω � Λ ( σ ) = Φ X ( σ Λ ω Λ c ) . X ∩ Λ � = ∅ 5 / 14

Gibbs measures Gibbs measure on the volume Λ with inverse temperature β > 0: 1 e − β H ω Λ ( σ Λ ω Λ c ) . µ ω β, Λ ( σ ) = Z ω β, Λ Standard partition function: e − β H ω � Λ ( σ Λ ω Λ c ) . Z ω β, Λ = σ ∈ A Λ Gibbs measure on the infinite volume with inverse temperature β > 0: Probability measure µ β such that, for every Λ ⋐ Z µ ω β, Λ ( f ) := E µ β ( f |F Λ c )( ω ) µ β -a.s. , for all F Λ -measurable f (DLR equation). 6 / 14

Infinite mass Gibbs measures Alphabet A = N . Irredutible transition matrix Σ A ⊆ N N . φ : Σ A → R measurable potential. 7 / 14

Infinite mass Gibbs measures Alphabet A = N . Irredutible transition matrix Σ A ⊆ N N . φ : Σ A → R measurable potential. Beltrán, Bissacot, E. - 2019+: A formal definition of the sigma-finite Gibbs measure µ with potential φ such that µ (Σ A ) = ∞ . Only exists under conditions of the shift and the potential. 7 / 14

Infinite mass Gibbs measures Alphabet A = N . Irredutible transition matrix Σ A ⊆ N N . φ : Σ A → R measurable potential. Beltrán, Bissacot, E. - 2019+: A formal definition of the sigma-finite Gibbs measure µ with potential φ such that µ (Σ A ) = ∞ . Only exists under conditions of the shift and the potential. Moreover: Existence of Σ A (Renewal shift) and β c > 0 such that, for positive recurrent weakly Hölder continuous βφ with sup φ < ∞ : 1 For β < β c , all βφ -Gibbs measure are finite. 2 For β > β c , there exists an infinite βφ -Gibbs measure. 7 / 14

Infinite mass Gibbs measures Alphabet A = N . Irredutible transition matrix Σ A ⊆ N N . φ : Σ A → R measurable potential. Beltrán, Bissacot, E. - 2019+: A formal definition of the sigma-finite Gibbs measure µ with potential φ such that µ (Σ A ) = ∞ . Only exists under conditions of the shift and the potential. Moreover: Existence of Σ A (Renewal shift) and β c > 0 such that, for positive recurrent weakly Hölder continuous βφ with sup φ < ∞ : 1 For β < β c , all βφ -Gibbs measure are finite. 2 For β > β c , there exists an infinite βφ -Gibbs measure. Charles Pfister at CIRM, Marseille, 2013: "We should consider infinite measures on Statistical Mechanics. People from Ergodic Theory already did it..." Classical reference: An Introduction to Infinite Ergodic Theory, 1997 by Jon Aaronson. 7 / 14

Motivation: Markov chains and Markov fields A is a finite set (e.g. { 0 , 1 } ). Ω = A Z is the set of configurations on Z . One-sided Markov chain: Probability measures on Ω on an infinite future conditioned on an one-step past. − n − 1 − n Two-sided Markov chain: Probability measures on Ω on a finite volume conditioned on an one-step past and future. n − n − n − 1 n + 1 Question: After n → ∞ , are one-sided and two sided Markov chain equivalent? YES (Brascamp, Spitzer) 8 / 14

g -measures and Gibbs measures g -measures: Probability measures on Ω on an infinite future conditioned on an infinite past. Generalization of Markov chain. − n Gibbs measures: Probability measures on Ω on a finite volume conditioned on an infinite outside (past and future). n − n Question: After n → ∞ , are g -measures and Gibbs measures equivalent? NO Fernández, Gallo, Maillard (ECP, 2011): g -measure - non-Gibbs measure. Bissacot, E., van Enter, Le Ny (CMP, 2018): Gibbs measure - non- g -measure. 9 / 14

Dyson model Consider A = {− 1 , + 1 } . Long range Ising model: For σ ∈ {− 1 , + 1 } Z , and 1 < α ≤ 2, J � H ( σ ) = − | i − j | α σ i σ j , i , j ∈ Z i � = j where J > 0 (Ferromagnetic). (Ising - Z. Phys.,1922) Nearest-neighbors Ising model on Z : No phase transition at any temperature. (Dyson - CMP,1969) Long-range Ising model on Z : Phase transition at low temperature. Phase Transition: High temperature - one Gibbs measure; low temperature - more than one Gibbs measures. 10 / 14

Result log 2 ∼ Define α + := 3 − log 3 = 1 . 415. Theorem (Bissacot, E., van Enter, Le Ny - CMP, 2018) For every Gibbs measure µ of the Dyson model with exponent α + < α < 2 at sufficiently low temperature, the one-sided conditional probability E µ ([ ω 0 ] |F < 0 ]( · ) is essentially discontinuous at ω alt = (( − 1 ) i ) i ∈ Z . ω alt + − + − + − + − + − + − + − + − + − + − + − + 0 11 / 14

Wetting transition and finite energy Wetting transition: For every L ≥ 1, there exists N ≥ 1 such that − phase − phase − − − − − − 0 − N − L − N − 1 L Finite energy: There exists c > 0 s.t., for every ω ∈ {− 1 , 1 } Z , � � � � J � � � | i − j | α ( − 1 ) i ω j � < c , for every L 1 ≥ 1 . � � � � j / ∈ [ − L 1 , − 1 ] i ∈ [ − L 1 , − 1 ] � � Weak enough to change the phase. J ij ( − 1 ) i ω j ω ω − + − + + − + − + − + 12 / 14

Sketch of the proof + phase + + + + + + + − + + + + + + + + + + + + + + − + − + − + − + − N − L 1 − L 1 0 − phase − − − − − − − − − − − + − + − + − + − + − N − L 1 − L 1 0 13 / 14

References E. Beltrán, R. Bissacot, E.O. Endo. Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts . (2019+). R. Bissacot, E.O. Endo, A.C.D. van Enter, A. Le Ny. Entropic repulsion and lack of the g -measure property for Dyson models. Comm. Math. Phys. 363 , 767–788, (2018). R. Fernández, S. Gallo, G. Maillard. Regular g -measures are not always Gibbsian. El. Comm. Prob. , 16 , 732–740, (2011). M. Keane. Strongly mixing g-measures. Inventiones Math. 16 , 309–24 (1972). O.M. Sarig. Lecture Notes on Thermodynamic Formalism for Topological Markov Shifts. Penn State (2009) 14 / 14