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. Endo1 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

1e-mail address: eoe2@nyu.edu

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g-function

A: finite set (e.g. {−1, 1}); Set of sequences (Full shift) AZ− = {(. . . , x−1, x0) : xn ∈ A}; T : AZ− → AZ− be the shift defined by T(. . . , x−1, x0) = (. . . , x−2, x−1). g-functions: Continuous positive functions g : AZ− → (0, 1) such that for all x ∈ AZ−,

• y∈T −1x

g(y) = 1.

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Ruelle Operator

Ruelle operator: For a fixed g-function, (Lgh)(x) =

• y∈T −1x

g(y)h(y), for all continuous function h on AZ−. Dual of the Ruelle Operator: Defined on the set of probability measures by

• AZ− h(x)L∗

gµ(d x) =

• AZ−(Lgh)(x)µ(d x)

for all continuous functions h on AZ−.

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g-measure

Definition A probability measure µ is a g-measure if there is a g-function g such that L∗

gµ = µ.

g-measure is translation-invariant. Can be extended to AZ. Ledrappier: µ is a g-measure if, and only if, for every ω0 ∈ A, Eµ [✶σ0=ω0|F<0] (ω) = g(ωZ−) µ − a.e. g-measure is also called Chain of Infinite Order. Markov chain given all past.

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Gibbs measures

Finite subset of Z: Λ ⋐ Z (Interval). Configuration: σΛ ∈ AΛ. Boundary condition: ωΛc ∈ AΛc. FΛ be the sigma algebra on Λ generated by the cylinder sets. Interaction: Collection Φ = (ΦX)X⋐Z of FX-measurable functions ΦX : AX → R

• ver all

X ⋐ Z. Hamiltonian: Sum of interactions Hω

Λ (σ) =

• X∩Λ=∅

ΦX(σΛωΛc).

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Gibbs measures

Gibbs measure on the volume Λ with inverse temperature β > 0: µω

β,Λ(σ) =

1 Z ω

β,Λ

e−βHω

Λ (σΛωΛc ).

Standard partition function: Z ω

β,Λ =

• σ∈AΛ

e−βHω

Λ (σΛωΛc ).

Gibbs measure on the infinite volume with inverse temperature β > 0: Probability measure µβ such that, for every Λ ⋐ Z µω

β,Λ(f ) := Eµβ(f |FΛc)(ω)

µβ-a.s., for all FΛ-measurable f (DLR equation).

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Infinite mass Gibbs measures

Alphabet A = N. Irredutible transition matrix ΣA ⊆ NN. φ : ΣA → R measurable potential.

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Infinite mass Gibbs measures

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

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Infinite mass Gibbs measures

Alphabet A = N. Irredutible transition matrix ΣA ⊆ NN. φ : ΣA → R measurable potential. Beltrán, Bissacot, E. - 2019+: A formal definition of the sigma-finite Gibbs measure µ with potential φ such that µ(ΣA) = ∞. Only exists under conditions of the shift and the potential. Moreover: Existence of ΣA (Renewal shift) and βc > 0 such that, for positive recurrent weakly Hölder continuous βφ with sup φ < ∞:

1 For β < βc, all βφ-Gibbs measure are finite. 2 For β > βc, there exists an infinite βφ-Gibbs measure.

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Infinite mass Gibbs measures

Alphabet A = N. Irredutible transition matrix ΣA ⊆ NN. φ : ΣA → R measurable potential. Beltrán, Bissacot, E. - 2019+: A formal definition of the sigma-finite Gibbs measure µ with potential φ such that µ(ΣA) = ∞. Only exists under conditions of the shift and the potential. Moreover: Existence of ΣA (Renewal shift) and βc > 0 such that, for positive recurrent weakly Hölder continuous βφ with sup φ < ∞:

1 For β < βc, all βφ-Gibbs measure are finite. 2 For β > βc, 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.

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Motivation: Markov chains and Markov fields

A is a finite set (e.g. {0, 1}). Ω = AZ 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 −n − 1 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)

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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.

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Dyson model

Consider A = {−1, +1}. Long range Ising model: For σ ∈ {−1, +1}Z, and 1 < α ≤ 2, H(σ) = −

• i,j∈Z

i=j

J |i − j|α σiσj, 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.

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Result

Define α+ := 3 − log 3

log 2 ∼

= 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 Eµ([ω0]|F<0](·) is essentially discontinuous at ωalt = ((−1)i)i∈Z. ωalt − − − − − − − − − − − + + + + + + + + + + + +

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Wetting transition and finite energy

Wetting transition: For every L ≥ 1, there exists N ≥ 1 such that −N − L −N − 1 L − phase − − − − − − − phase Finite energy: There exists c > 0 s.t., for every ω ∈ {−1, 1}Z,

• j /

∈[−L1,−1]

• i∈[−L1,−1]

J |i − j|α (−1)iωj

• < c,

for every L1 ≥ 1. Weak enough to change the phase.

Jij(−1)iωj

ω ω − − − − − + + + + + +

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Sketch of the proof

−N − L1 −L1 + + + + + + + + + + − − − − − + + + + + + + + + + + + + + + + phase −N − L1 −L1 − − − − − − − − − − − − − − − + + + + + − phase

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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)

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