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Gibbs-non-Gibbs dynamical transitions. A large-deviation paradigm - - PowerPoint PPT Presentation

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Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Gibbs-non-Gibbs dynamical transitions. A large-deviation paradigm R. Fern andez F. den Hollander J. Mart nez Utrecht Leiden Leiden Kyoto, July 2013 Intro Gibbs Non-Gibbs

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Intro Gibbs Non-Gibbs Dynamics Mean field Kac End

Gibbs-non-Gibbs dynamical transitions. A large-deviation paradigm

  • R. Fern´

andez

  • F. den Hollander
  • J. Mart´

ınez

Utrecht Leiden Leiden

Kyoto, July 2013

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SLIDE 2

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Gibbs measures and their transformations

The Gibbs – non-Gibbs saga

Intuitively, µ Gibbs if µ ∝ e−βH This is, however, valid only on finite regions To pass to the thermodynamic limit must introduce:

◮ Interactions ◮ Specifications

Gibbs measures designed to describe equilibrium No reason to expect them out of equilibrium, e.g. under evolutions

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SLIDE 3

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Gibbs measures and their transformations

The Gibbs – non-Gibbs saga

Intuitively, µ Gibbs if µ ∝ e−βH This is, however, valid only on finite regions To pass to the thermodynamic limit must introduce:

◮ Interactions ◮ Specifications

Gibbs measures designed to describe equilibrium No reason to expect them out of equilibrium, e.g. under evolutions

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SLIDE 4

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Gibbs measures and their transformations

The Gibbs – non-Gibbs saga

Intuitively, µ Gibbs if µ ∝ e−βH This is, however, valid only on finite regions To pass to the thermodynamic limit must introduce:

◮ Interactions ◮ Specifications

Gibbs measures designed to describe equilibrium No reason to expect them out of equilibrium, e.g. under evolutions

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SLIDE 5

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Gibbs measures and their transformations

Examples of non-Gibsianness

Renormalization transformations

◮ Block-renormalization: blocks of spins → effective spins ◮ Renormalized measure: coarser, blurred ◮ In many instances: renormalized measure non-Gibbsian ◮ Reason: hidden variables bringing info from infinite

Stochastic evolutions Gibbs measures subjected to Glauber dynamics

◮ Can loose Gibsianness at some finite time ◮ Gibbsianness recovered in some cases

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Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Gibbs measures and their transformations

Examples of non-Gibsianness

Renormalization transformations

◮ Block-renormalization: blocks of spins → effective spins ◮ Renormalized measure: coarser, blurred ◮ In many instances: renormalized measure non-Gibbsian ◮ Reason: hidden variables bringing info from infinite

Stochastic evolutions Gibbs measures subjected to Glauber dynamics

◮ Can loose Gibsianness at some finite time ◮ Gibbsianness recovered in some cases

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SLIDE 7

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Gibbs measures and their transformations

Examples of non-Gibsianness

Renormalization transformations

◮ Block-renormalization: blocks of spins → effective spins ◮ Renormalized measure: coarser, blurred ◮ In many instances: renormalized measure non-Gibbsian ◮ Reason: hidden variables bringing info from infinite

Stochastic evolutions Gibbs measures subjected to Glauber dynamics

◮ Can loose Gibsianness at some finite time ◮ Gibbsianness recovered in some cases

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SLIDE 8

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Gibbs measures and their transformations

Dynamic non-Gibbsianness

Original explanation

◮ Two-slice system: past acts as hidden variables for present ◮ Two-slice system ∼ equilibrium duplicated variables

Alternative paradigm

◮ Intuitively: most probable history of an improbable state ◮ Formally: large deviations in trajectory space ◮ Non-Gibbs = multiple optimal trajectories → discontinuity

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Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Gibbs measures and their transformations

Dynamic non-Gibbsianness

Original explanation

◮ Two-slice system: past acts as hidden variables for present ◮ Two-slice system ∼ equilibrium duplicated variables

Alternative paradigm

◮ Intuitively: most probable history of an improbable state ◮ Formally: large deviations in trajectory space ◮ Non-Gibbs = multiple optimal trajectories → discontinuity

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SLIDE 10

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Gibbs measures and their transformations

Dynamic non-Gibbsianness

Original explanation

◮ Two-slice system: past acts as hidden variables for present ◮ Two-slice system ∼ equilibrium duplicated variables

Alternative paradigm

◮ Intuitively: most probable history of an improbable state ◮ Formally: large deviations in trajectory space ◮ Non-Gibbs = multiple optimal trajectories → discontinuity

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Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Plan

Plan and credits

In this talk:

◮ Review of Gibbsianness ◮ Review of original proof of dynamical non-Gibbsianness ◮ New paradigm for dynamical non-Gibbsianness ◮ Rigorous results for

◮ Mean-field spin models ◮ Kac models

Acknowledgemens: Aernout van Enter (Groningen) Franck Redig (Delft) Christoff K¨ ulske (Bochum) Victor Ermolaev (Nedap)

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Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Plan

Plan and credits

In this talk:

◮ Review of Gibbsianness ◮ Review of original proof of dynamical non-Gibbsianness ◮ New paradigm for dynamical non-Gibbsianness ◮ Rigorous results for

◮ Mean-field spin models ◮ Kac models

Acknowledgemens: Aernout van Enter (Groningen) Franck Redig (Delft) Christoff K¨ ulske (Bochum) Victor Ermolaev (Nedap)

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Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Plan

Plan and credits

In this talk:

◮ Review of Gibbsianness ◮ Review of original proof of dynamical non-Gibbsianness ◮ New paradigm for dynamical non-Gibbsianness ◮ Rigorous results for

◮ Mean-field spin models ◮ Kac models

Acknowledgemens: Aernout van Enter (Groningen) Franck Redig (Delft) Christoff K¨ ulske (Bochum) Victor Ermolaev (Nedap)

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SLIDE 14

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Definition

Lattice systems

Basic ingredients:

◮ Lattice L: e.g. Zd ◮ Single-spin space S: e.g. {−1, 1} ◮ Configuration space Ω = SL

Topology and σ-algebra F generated by cylinders: Cω

Λ =

  • ω ∈ Ω : ω

Λ = σ Λ

  • , Λ ⊂

⊂ L

  • ω

Λ = (ωx)x∈Λ

  • Interaction: Family of local functions (=local contributions)

Φ =

  • φB : Ω → R , FB−measurable
  • [φB(ω) = φB(σ) if ω

B = σ B]

Hamiltonian on region Λ given σ outside: HΛ(ω | σ) =

  • B:B∩Λ=∅

φB(ωΛσ)

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Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Definition

Lattice systems

Basic ingredients:

◮ Lattice L: e.g. Zd ◮ Single-spin space S: e.g. {−1, 1} ◮ Configuration space Ω = SL

Topology and σ-algebra F generated by cylinders: Cω

Λ =

  • ω ∈ Ω : ω

Λ = σ Λ

  • , Λ ⊂

⊂ L

  • ω

Λ = (ωx)x∈Λ

  • Interaction: Family of local functions (=local contributions)

Φ =

  • φB : Ω → R , FB−measurable
  • [φB(ω) = φB(σ) if ω

B = σ B]

Hamiltonian on region Λ given σ outside: HΛ(ω | σ) =

  • B:B∩Λ=∅

φB(ωΛσ)

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SLIDE 16

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Definition

Lattice systems

Basic ingredients:

◮ Lattice L: e.g. Zd ◮ Single-spin space S: e.g. {−1, 1} ◮ Configuration space Ω = SL

Topology and σ-algebra F generated by cylinders: Cω

Λ =

  • ω ∈ Ω : ω

Λ = σ Λ

  • , Λ ⊂

⊂ L

  • ω

Λ = (ωx)x∈Λ

  • Interaction: Family of local functions (=local contributions)

Φ =

  • φB : Ω → R , FB−measurable
  • [φB(ω) = φB(σ) if ω

B = σ B]

Hamiltonian on region Λ given σ outside: HΛ(ω | σ) =

  • B:B∩Λ=∅

φB(ωΛσ)

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SLIDE 17

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Definition

Lattice systems

Basic ingredients:

◮ Lattice L: e.g. Zd ◮ Single-spin space S: e.g. {−1, 1} ◮ Configuration space Ω = SL

Topology and σ-algebra F generated by cylinders: Cω

Λ =

  • ω ∈ Ω : ω

Λ = σ Λ

  • , Λ ⊂

⊂ L

  • ω

Λ = (ωx)x∈Λ

  • Interaction: Family of local functions (=local contributions)

Φ =

  • φB : Ω → R , FB−measurable
  • [φB(ω) = φB(σ) if ω

B = σ B]

Hamiltonian on region Λ given σ outside: HΛ(ω | σ) =

  • B:B∩Λ=∅

φB(ωΛσ)

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Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Definition

(Lattice) Gibbs measures: formal definition

Gibbsian specification: Family ΠΦ = {πΦ

Λ : Λ ⊂

⊂ L} with πΦ

Λ(Cω

Λ) = e−βHΛ(ω|σ)

Norm. [πΦ

Λ( · | σ) = equilibrium in Λ given σ]

Gibbs measures: µ is Gibbs for Φ if, equivalently,

◮ µ is left invariant by ΠΦ:

  • πΦ

Λ(Cω

Λ) µ(dω) = µ(Cω Λ)

[µ = equilibrium in L = every Λ in equilibrium]

◮ µ = w − limΛ→L πΦ Λ(· | σ) + convex combinations

[thermodynamic limit]

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Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Definition

(Lattice) Gibbs measures: formal definition

Gibbsian specification: Family ΠΦ = {πΦ

Λ : Λ ⊂

⊂ L} with πΦ

Λ(Cω

Λ) = e−βHΛ(ω|σ)

Norm. [πΦ

Λ( · | σ) = equilibrium in Λ given σ]

Gibbs measures: µ is Gibbs for Φ if, equivalently,

◮ µ is left invariant by ΠΦ:

  • πΦ

Λ(Cω

Λ) µ(dω) = µ(Cω Λ)

[µ = equilibrium in L = every Λ in equilibrium]

◮ µ = w − limΛ→L πΦ Λ(· | σ) + convex combinations

[thermodynamic limit]

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Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Definition

(Lattice) Gibbs measures: formal definition

Gibbsian specification: Family ΠΦ = {πΦ

Λ : Λ ⊂

⊂ L} with πΦ

Λ(Cω

Λ) = e−βHΛ(ω|σ)

Norm. [πΦ

Λ( · | σ) = equilibrium in Λ given σ]

Gibbs measures: µ is Gibbs for Φ if, equivalently,

◮ µ is left invariant by ΠΦ:

  • πΦ

Λ(Cω

Λ) µ(dω) = µ(Cω Λ)

[µ = equilibrium in L = every Λ in equilibrium]

◮ µ = w − limΛ→L πΦ Λ(· | σ) + convex combinations

[thermodynamic limit]

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Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Gibbsianness test

How to recognize Gibbsianness

Kozlov – Sullivan: µ is Gibbs iff it is

◮ Non-null: µ(Cω

Λ) > 0 for every cylinder Cω Λ

◮ Quasilocal: If Λ ⊂ Γ ⊂

⊂ L, sup

σ,ω,ξ±

  • µ
  • Cω

Λ

  • σ

Γ ξ+

− µ

  • Cω

Λ

  • σ

Γ ξ−

  • −

− − →

γ→L 0

Physics in Λ does not depend on state of the Andromeda galaxy

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Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Gibbsianness test

How to recognize Gibbsianness

Kozlov – Sullivan: µ is Gibbs iff it is

◮ Non-null: µ(Cω

Λ) > 0 for every cylinder Cω Λ

◮ Quasilocal: If Λ ⊂ Γ ⊂

⊂ L, sup

σ,ω,ξ±

  • µ
  • Cω

Λ

  • σ

Γ ξ+

− µ

  • Cω

Λ

  • σ

Γ ξ−

  • −

− − →

γ→L 0

Physics in Λ does not depend on state of the Andromeda galaxy

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Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Gibbsianness test

How to recognize Gibbsianness

Kozlov – Sullivan: µ is Gibbs iff it is

◮ Non-null: µ(Cω

Λ) > 0 for every cylinder Cω Λ

◮ Quasilocal: If Λ ⊂ Γ ⊂

⊂ L, sup

σ,ω,ξ±

  • µ
  • Cω

Λ

  • σ

Γ ξ+

− µ

  • Cω

Λ

  • σ

Γ ξ−

  • −

− − →

γ→L 0

Physics in Λ does not depend on state of the Andromeda galaxy

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Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Non-quasilocality

Essential non-quasilocality

µ is Gibbs if ∃ Λ and wsp s.t: ∃ ξ± for which sup

σ,ω,ξ±

  • µ
  • Cω

Λ

  • σ

Γ ξ+

− µ

  • Cω

Λ

  • σ

Γ ξ−

  • —

/ − →

γ→L 0

for every realisation of µ

  • Cω

Λ

  • ·
  • ◮ Quasilocality = continuity w.r.t. external conditions

◮ Non-quasilocality = essential discontinuity w.r.t. external

conditions Interpretation: Info from ∞ despite frozen fluctuations Possible explanation: hidden variables

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Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Non-quasilocality

Essential non-quasilocality

µ is Gibbs if ∃ Λ and wsp s.t: ∃ ξ± for which sup

σ,ω,ξ±

  • µ
  • Cω

Λ

  • σ

Γ ξ+

− µ

  • Cω

Λ

  • σ

Γ ξ−

  • —

/ − →

γ→L 0

for every realisation of µ

  • Cω

Λ

  • ·
  • ◮ Quasilocality = continuity w.r.t. external conditions

◮ Non-quasilocality = essential discontinuity w.r.t. external

conditions Interpretation: Info from ∞ despite frozen fluctuations Possible explanation: hidden variables

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SLIDE 26

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Non-quasilocality

Essential non-quasilocality

µ is Gibbs if ∃ Λ and wsp s.t: ∃ ξ± for which sup

σ,ω,ξ±

  • µ
  • Cω

Λ

  • σ

Γ ξ+

− µ

  • Cω

Λ

  • σ

Γ ξ−

  • —

/ − →

γ→L 0

for every realisation of µ

  • Cω

Λ

  • ·
  • ◮ Quasilocality = continuity w.r.t. external conditions

◮ Non-quasilocality = essential discontinuity w.r.t. external

conditions Interpretation: Info from ∞ despite frozen fluctuations Possible explanation: hidden variables

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SLIDE 27

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Non-quasilocality

Essential non-quasilocality

µ is Gibbs if ∃ Λ and wsp s.t: ∃ ξ± for which sup

σ,ω,ξ±

  • µ
  • Cω

Λ

  • σ

Γ ξ+

− µ

  • Cω

Λ

  • σ

Γ ξ−

  • —

/ − →

γ→L 0

for every realisation of µ

  • Cω

Λ

  • ·
  • ◮ Quasilocality = continuity w.r.t. external conditions

◮ Non-quasilocality = essential discontinuity w.r.t. external

conditions Interpretation: Info from ∞ despite frozen fluctuations Possible explanation: hidden variables

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Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness

Renormalization transformations

General definition: A (stochastic) RT is a map Prob(Ω) − → Prob(Ω′) µ − → µ′( · ) =

  • K( · | ω) µ(dω)

where K is a probability kernel The transformation is deterministic if ∃ f : Ω → Ω′ s.t. K( · | ω) = δ

f(ω)( · )

A block RT is of the form K(dω′ | ω) =

  • x′

K′

x(dω′ x′ | ω

Bx′ )

each Bx′ ⊂ ⊂ L is the block associated to x′

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Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness

Renormalization transformations

General definition: A (stochastic) RT is a map Prob(Ω) − → Prob(Ω′) µ − → µ′( · ) =

  • K( · | ω) µ(dω)

where K is a probability kernel The transformation is deterministic if ∃ f : Ω → Ω′ s.t. K( · | ω) = δ

f(ω)( · )

A block RT is of the form K(dω′ | ω) =

  • x′

K′

x(dω′ x′ | ω

Bx′ )

each Bx′ ⊂ ⊂ L is the block associated to x′

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Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness

Renormalization transformations

General definition: A (stochastic) RT is a map Prob(Ω) − → Prob(Ω′) µ − → µ′( · ) =

  • K( · | ω) µ(dω)

where K is a probability kernel The transformation is deterministic if ∃ f : Ω → Ω′ s.t. K( · | ω) = δ

f(ω)( · )

A block RT is of the form K(dω′ | ω) =

  • x′

K′

x(dω′ x′ | ω

Bx′ )

each Bx′ ⊂ ⊂ L is the block associated to x′

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Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness

Examples of block transformations

Deterministic transformations:

◮ Decimation ◮ Majority (odd block)

Stochastic transformations:

◮ Majority (even block) ◮ Kadanoff:

K′

x(dω′ x′ | ω

Bx′ ) =

exp

  • p ω′

x′

  • x∈Bx′ ωx
  • Norm

dω′

x′

[weighted majority; → majority as p → ∞]

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Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness

Hidden variables and non-quasilocality

Hidden-variables mechanism:

◮ Each fixed ω′

Λc determines a constrained Ω system

◮ ω′sp is s.t. the constrained system has a phase transition ◮ ξ′ far away decides the phase → info form ∞

Two-slice point of view:

◮ Ω = original slice = hidden variables ◮ Ω′ = present slice = observed variables

Non-quasilocality: Hidden variables constrained by observed variables exhibit a phase transition

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Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness

Hidden variables and non-quasilocality

Hidden-variables mechanism:

◮ Each fixed ω′

Λc determines a constrained Ω system

◮ ω′sp is s.t. the constrained system has a phase transition ◮ ξ′ far away decides the phase → info form ∞

Two-slice point of view:

◮ Ω = original slice = hidden variables ◮ Ω′ = present slice = observed variables

Non-quasilocality: Hidden variables constrained by observed variables exhibit a phase transition

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Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness

Hidden variables and non-quasilocality

Hidden-variables mechanism:

◮ Each fixed ω′

Λc determines a constrained Ω system

◮ ω′sp is s.t. the constrained system has a phase transition ◮ ξ′ far away decides the phase → info form ∞

Two-slice point of view:

◮ Ω = original slice = hidden variables ◮ Ω′ = present slice = observed variables

Non-quasilocality: Hidden variables constrained by observed variables exhibit a phase transition

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SLIDE 35

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness

Hidden variables and non-quasilocality

Hidden-variables mechanism:

◮ Each fixed ω′

Λc determines a constrained Ω system

◮ ω′sp is s.t. the constrained system has a phase transition ◮ ξ′ far away decides the phase → info form ∞

Two-slice point of view:

◮ Ω = original slice = hidden variables ◮ Ω′ = present slice = observed variables

Non-quasilocality: Hidden variables constrained by observed variables exhibit a phase transition

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SLIDE 36

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness

Single-site Kadanoff transformations (Bx′ = {x′})

On finite volumes, the two-slice measures are of the form KΛ(dω′ | ω) µ

Λ(dω) ∝ exp

  • β
  • HKad

Λ

(ω, ω′) + HΛ(ω)

  • dω′

Λ dω Λ

where HKad

Λ

(ω, ω′) =

  • x′

p β ω′

xωx − 1

β log

  • 2 cosh(p ωx)
  • acts on the original spins as an extra magnetic field

Constrained internal spins have phase transition if p β ω′

x compensates h in the average

(∗) and β is large enough

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SLIDE 37

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness

Single-site Kadanoff transformations (Bx′ = {x′})

On finite volumes, the two-slice measures are of the form KΛ(dω′ | ω) µ

Λ(dω) ∝ exp

  • β
  • HKad

Λ

(ω, ω′) + HΛ(ω)

  • dω′

Λ dω Λ

where HKad

Λ

(ω, ω′) =

  • x′

p β ω′

xωx − 1

β log

  • 2 cosh(p ωx)
  • acts on the original spins as an extra magnetic field

Constrained internal spins have phase transition if p β ω′

x compensates h in the average

(∗) and β is large enough

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SLIDE 38

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness

Single-site Kadanoff transformations (Bx′ = {x′})

On finite volumes, the two-slice measures are of the form KΛ(dω′ | ω) µ

Λ(dω) ∝ exp

  • β
  • HKad

Λ

(ω, ω′) + HΛ(ω)

  • dω′

Λ dω Λ

where HKad

Λ

(ω, ω′) =

  • x′

p β ω′

xωx − 1

β log

  • 2 cosh(p ωx)
  • acts on the original spins as an extra magnetic field

Constrained internal spins have phase transition if p β ω′

x compensates h in the average

(∗) and β is large enough

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SLIDE 39

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness

Single-site Kadanoff transformations (cont.)

Let h be the original Ising field and fix β large enough s.t.

◮ Original model with h = 0 has phase transition ◮ Pirogov-Sinai theory holds

If h = 0, alternated ω′ = ⇒ (∗) for p/β small enough Hence, ∃ p1 > p2 s.t.

◮ µ′ is Gibbs for p > p1 ◮ µ′ is not Gibbs for 2 > p

If h = 0, ∃ ω′ s.t. (∗) only for a range of p/β Hence, ∃ p1 ≥ p2 > p3 ≥ p4 s.t.

◮ µ′ is Gibbs for p > p1 ◮ µ′ is not Gibbs for p2 > p > p3 ◮ µ′ is not Gibbs for p4 > p

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Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness

Single-site Kadanoff transformations (cont.)

Let h be the original Ising field and fix β large enough s.t.

◮ Original model with h = 0 has phase transition ◮ Pirogov-Sinai theory holds

If h = 0, alternated ω′ = ⇒ (∗) for p/β small enough Hence, ∃ p1 > p2 s.t.

◮ µ′ is Gibbs for p > p1 ◮ µ′ is not Gibbs for 2 > p

If h = 0, ∃ ω′ s.t. (∗) only for a range of p/β Hence, ∃ p1 ≥ p2 > p3 ≥ p4 s.t.

◮ µ′ is Gibbs for p > p1 ◮ µ′ is not Gibbs for p2 > p > p3 ◮ µ′ is not Gibbs for p4 > p

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SLIDE 41

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness

Single-site Kadanoff transformations (cont.)

Let h be the original Ising field and fix β large enough s.t.

◮ Original model with h = 0 has phase transition ◮ Pirogov-Sinai theory holds

If h = 0, alternated ω′ = ⇒ (∗) for p/β small enough Hence, ∃ p1 > p2 s.t.

◮ µ′ is Gibbs for p > p1 ◮ µ′ is not Gibbs for 2 > p

If h = 0, ∃ ω′ s.t. (∗) only for a range of p/β Hence, ∃ p1 ≥ p2 > p3 ≥ p4 s.t.

◮ µ′ is Gibbs for p > p1 ◮ µ′ is not Gibbs for p2 > p > p3 ◮ µ′ is not Gibbs for p4 > p

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SLIDE 42

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness

Single-site Kadanoff transformations (cont.)

Let h be the original Ising field and fix β large enough s.t.

◮ Original model with h = 0 has phase transition ◮ Pirogov-Sinai theory holds

If h = 0, alternated ω′ = ⇒ (∗) for p/β small enough Hence, ∃ p1 > p2 s.t.

◮ µ′ is Gibbs for p > p1 ◮ µ′ is not Gibbs for 2 > p

If h = 0, ∃ ω′ s.t. (∗) only for a range of p/β Hence, ∃ p1 ≥ p2 > p3 ≥ p4 s.t.

◮ µ′ is Gibbs for p > p1 ◮ µ′ is not Gibbs for p2 > p > p3 ◮ µ′ is not Gibbs for p4 > p

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Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness

Single-site Kadanoff transformations (cont.)

Let h be the original Ising field and fix β large enough s.t.

◮ Original model with h = 0 has phase transition ◮ Pirogov-Sinai theory holds

If h = 0, alternated ω′ = ⇒ (∗) for p/β small enough Hence, ∃ p1 > p2 s.t.

◮ µ′ is Gibbs for p > p1 ◮ µ′ is not Gibbs for 2 > p

If h = 0, ∃ ω′ s.t. (∗) only for a range of p/β Hence, ∃ p1 ≥ p2 > p3 ≥ p4 s.t.

◮ µ′ is Gibbs for p > p1 ◮ µ′ is not Gibbs for p2 > p > p3 ◮ µ′ is not Gibbs for p4 > p

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SLIDE 44

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Definition and example

Dynamic Gibbs – non-Gibbs transitions

Simulations, particle systems, PCA: µt = Stµ with St= semigroup of operators. Dynamic G–non-G: µ Gibbs but µt non-Gibbs at some t Example:

◮ µ=low-T Ising model ◮ St = Sn infinite-T discrete-time Glauber

S =

  • x

S{x} with Sx(ωx | ωx) = 1 − ǫ Sx(−ωx | ωx) = ǫ [invariant measure = product measure = infinite-T Gibbs] Unquenching: heating up a low-T Ising model

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SLIDE 45

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Definition and example

Dynamic Gibbs – non-Gibbs transitions

Simulations, particle systems, PCA: µt = Stµ with St= semigroup of operators. Dynamic G–non-G: µ Gibbs but µt non-Gibbs at some t Example:

◮ µ=low-T Ising model ◮ St = Sn infinite-T discrete-time Glauber

S =

  • x

S{x} with Sx(ωx | ωx) = 1 − ǫ Sx(−ωx | ωx) = ǫ [invariant measure = product measure = infinite-T Gibbs] Unquenching: heating up a low-T Ising model

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SLIDE 46

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Definition and example

Dynamic Gibbs – non-Gibbs transitions

Simulations, particle systems, PCA: µt = Stµ with St= semigroup of operators. Dynamic G–non-G: µ Gibbs but µt non-Gibbs at some t Example:

◮ µ=low-T Ising model ◮ St = Sn infinite-T discrete-time Glauber

S =

  • x

S{x} with Sx(ωx | ωx) = 1 − ǫ Sx(−ωx | ωx) = ǫ [invariant measure = product measure = infinite-T Gibbs] Unquenching: heating up a low-T Ising model

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SLIDE 47

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Definition and example

Dynamic Gibbs – non-Gibbs transitions

Simulations, particle systems, PCA: µt = Stµ with St= semigroup of operators. Dynamic G–non-G: µ Gibbs but µt non-Gibbs at some t Example:

◮ µ=low-T Ising model ◮ St = Sn infinite-T discrete-time Glauber

S =

  • x

S{x} with Sx(ωx | ωx) = 1 − ǫ Sx(−ωx | ωx) = ǫ [invariant measure = product measure = infinite-T Gibbs] Unquenching: heating up a low-T Ising model

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SLIDE 48

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Definition and example

Un-quenching G–non-G transitions

In matrix form

  • Sx
  • ω ω′ ≡ Sx(ω′

x | ωx)

Sx = 1 − ǫ ǫ ǫ 1 − ǫ

  • ,

Sn

x = 1

2 1 + an 1 − an 1 − an 1 + an

  • with an = (1 − 2ǫ)n. Hence

Sn

x(ω′ x | ωx) = An epnω′

xωx

, pn = log 1 + an 1 − an

  • Kadanoff with pn−

− − →

n→0 ∞ and pn−

− − →

n→∞ 0

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SLIDE 49

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Definition and example

Un-quenching G–non-G transitions

In matrix form

  • Sx
  • ω ω′ ≡ Sx(ω′

x | ωx)

Sx = 1 − ǫ ǫ ǫ 1 − ǫ

  • ,

Sn

x = 1

2 1 + an 1 − an 1 − an 1 + an

  • with an = (1 − 2ǫ)n. Hence

Sn

x(ω′ x | ωx) = An epnω′

xωx

, pn = log 1 + an 1 − an

  • Kadanoff with pn−

− − →

n→0 ∞ and pn−

− − →

n→∞ 0

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SLIDE 50

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Definition and example

Un-quenching G–non-G transitions (cont.)

Using previous results on Kadanoff-renormalized measures: (h = 0) Gibbs n1 . . . . . .

✲

Non-Gibbs n2 (h > 0) Gibbs n1 . . . n2 Non-Gibbs n3 . . .

✲

Gibbs n4 Mathematical mechanism: hidden variables (two-slice view) Physical mechanism?

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SLIDE 51

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Definition and example

Un-quenching G–non-G transitions (cont.)

Using previous results on Kadanoff-renormalized measures: (h = 0) Gibbs n1 . . . . . .

✲

Non-Gibbs n2 (h > 0) Gibbs n1 . . . n2 Non-Gibbs n3 . . .

✲

Gibbs n4 Mathematical mechanism: hidden variables (two-slice view) Physical mechanism?

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SLIDE 52

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Definition and example

Un-quenching G–non-G transitions (cont.)

Using previous results on Kadanoff-renormalized measures: (h = 0) Gibbs n1 . . . . . .

✲

Non-Gibbs n2 (h > 0) Gibbs n1 . . . n2 Non-Gibbs n3 . . .

✲

Gibbs n4 Mathematical mechanism: hidden variables (two-slice view) Physical mechanism?

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SLIDE 53

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mechanism

Alternative paradigm: Heuristic version

  • A. van Enter: most probable history of an improbable state

Given a large improbable droplet. How did it get there?

◮ Nurture: Created by the dynamics (cost exp-volume) ◮ Nature: Present at t = 0 and survived

To compete: typical of the other phase (cost exp-perimeter) Heuristic version:

◮ Short t: Only nature, no time to change much ◮ Mid t:

◮ ωsp nurtured, but ξ± nature ◮ Hence ξ± determines original phase → discontinuity

◮ Long t: If h = 0 only one phase → no tilting mechanism

slide-54
SLIDE 54

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mechanism

Alternative paradigm: Heuristic version

  • A. van Enter: most probable history of an improbable state

Given a large improbable droplet. How did it get there?

◮ Nurture: Created by the dynamics (cost exp-volume) ◮ Nature: Present at t = 0 and survived

To compete: typical of the other phase (cost exp-perimeter) Heuristic version:

◮ Short t: Only nature, no time to change much ◮ Mid t:

◮ ωsp nurtured, but ξ± nature ◮ Hence ξ± determines original phase → discontinuity

◮ Long t: If h = 0 only one phase → no tilting mechanism

slide-55
SLIDE 55

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mechanism

Alternative paradigm: Heuristic version

  • A. van Enter: most probable history of an improbable state

Given a large improbable droplet. How did it get there?

◮ Nurture: Created by the dynamics (cost exp-volume) ◮ Nature: Present at t = 0 and survived

To compete: typical of the other phase (cost exp-perimeter) Heuristic version:

◮ Short t: Only nature, no time to change much ◮ Mid t:

◮ ωsp nurtured, but ξ± nature ◮ Hence ξ± determines original phase → discontinuity

◮ Long t: If h = 0 only one phase → no tilting mechanism

slide-56
SLIDE 56

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mechanism

Alternative paradigm: Heuristic version

  • A. van Enter: most probable history of an improbable state

Given a large improbable droplet. How did it get there?

◮ Nurture: Created by the dynamics (cost exp-volume) ◮ Nature: Present at t = 0 and survived

To compete: typical of the other phase (cost exp-perimeter) Heuristic version:

◮ Short t: Only nature, no time to change much ◮ Mid t:

◮ ωsp nurtured, but ξ± nature ◮ Hence ξ± determines original phase → discontinuity

◮ Long t: If h = 0 only one phase → no tilting mechanism

slide-57
SLIDE 57

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mechanism

Alternative paradigm: Heuristic version

  • A. van Enter: most probable history of an improbable state

Given a large improbable droplet. How did it get there?

◮ Nurture: Created by the dynamics (cost exp-volume) ◮ Nature: Present at t = 0 and survived

To compete: typical of the other phase (cost exp-perimeter) Heuristic version:

◮ Short t: Only nature, no time to change much ◮ Mid t:

◮ ωsp nurtured, but ξ± nature ◮ Hence ξ± determines original phase → discontinuity

◮ Long t: If h = 0 only one phase → no tilting mechanism

slide-58
SLIDE 58

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mechanism

Alternative paradigm: Heuristic version

  • A. van Enter: most probable history of an improbable state

Given a large improbable droplet. How did it get there?

◮ Nurture: Created by the dynamics (cost exp-volume) ◮ Nature: Present at t = 0 and survived

To compete: typical of the other phase (cost exp-perimeter) Heuristic version:

◮ Short t: Only nature, no time to change much ◮ Mid t:

◮ ωsp nurtured, but ξ± nature ◮ Hence ξ± determines original phase → discontinuity

◮ Long t: If h = 0 only one phase → no tilting mechanism

slide-59
SLIDE 59

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mechanism

Alternative paradigm: rigorous version

most probable history = most probable trajectory most probable = minimizer of the large-deviation rate Paradigm: Establish a large-deviation principle for trajectories

  • f measures conditioned to a given final empirical measure

◮ Single minimizer = Gibbsianness ◮ Multiple minimizers = non-Gibbsianness

Perturbation of conditioning → discontinuous choice of trajectory

slide-60
SLIDE 60

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mechanism

Alternative paradigm: rigorous version

most probable history = most probable trajectory most probable = minimizer of the large-deviation rate Paradigm: Establish a large-deviation principle for trajectories

  • f measures conditioned to a given final empirical measure

◮ Single minimizer = Gibbsianness ◮ Multiple minimizers = non-Gibbsianness

Perturbation of conditioning → discontinuous choice of trajectory

slide-61
SLIDE 61

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mechanism

Alternative paradigm: rigorous version

most probable history = most probable trajectory most probable = minimizer of the large-deviation rate Paradigm: Establish a large-deviation principle for trajectories

  • f measures conditioned to a given final empirical measure

◮ Single minimizer = Gibbsianness ◮ Multiple minimizers = non-Gibbsianness

Perturbation of conditioning → discontinuous choice of trajectory

slide-62
SLIDE 62

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mechanism

Alternative paradigm: rigorous version

most probable history = most probable trajectory most probable = minimizer of the large-deviation rate Paradigm: Establish a large-deviation principle for trajectories

  • f measures conditioned to a given final empirical measure

◮ Single minimizer = Gibbsianness ◮ Multiple minimizers = non-Gibbsianness

Perturbation of conditioning → discontinuous choice of trajectory

slide-63
SLIDE 63

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mechanism

Alternative paradigm: graphical summary

[h = 0]

t1

  • c

m(t,) ^ t m(t,) ^ t

  • c

m*

  • m*

Forbidden Region

One trajectory = Gibbs Many trajectories = non-Gibbs

slide-64
SLIDE 64

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Definition

The program

Prove rigorously the previous paradigm. Steps: (i) Mean-field models (ii) Kac models (iii) Finite-range models At present: (i) and (ii) for Ising under independent dynamics (i) Mean-field:

◮ No geometry – no notion of neighbourhood ◮ Everything in terms of empirical magnetization

slide-65
SLIDE 65

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Definition

The program

Prove rigorously the previous paradigm. Steps: (i) Mean-field models (ii) Kac models (iii) Finite-range models At present: (i) and (ii) for Ising under independent dynamics (i) Mean-field:

◮ No geometry – no notion of neighbourhood ◮ Everything in terms of empirical magnetization

slide-66
SLIDE 66

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Definition

Mean-field Ising model

N Ising spins (ωi ∈ {−1, 1}) HN(σ) = − J

2N N

  • i,j=1

σiσj − h

N

  • i=1

σi = NH(mN(σ)) where mN is the empirical magnetization mN(σ) = 1 N

  • i=1

Nσi and, if m ∈ MN := {−1, −1 + 2N−1, . . . , +1 − 2N−1, +1}, H(m) := − 1

2Jm2 − hm

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SLIDE 67

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Definition

Mean-field Ising model

N Ising spins (ωi ∈ {−1, 1}) HN(σ) = − J

2N N

  • i,j=1

σiσj − h

N

  • i=1

σi = NH(mN(σ)) where mN is the empirical magnetization mN(σ) = 1 N

  • i=1

Nσi and, if m ∈ MN := {−1, −1 + 2N−1, . . . , +1 − 2N−1, +1}, H(m) := − 1

2Jm2 − hm

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SLIDE 68

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Definition

Mean-field measures and evolution

The HN-Gibbs measure [β absorbed] µN(dσ) = e−HN(σ) ZN dσ induces a measure on MN µN(dm) :=

  • N

1+m 2

N e−NH(m) Z

N

dm [µN ← → µN + permutation invariance]

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SLIDE 69

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Definition

Mean-field measures and evolution

The HN-Gibbs measure [β absorbed] µN(dσ) = e−HN(σ) ZN dσ induces a measure on MN µN(dm) :=

  • N

1+m 2

N e−NH(m) Z

N

dm [µN ← → µN + permutation invariance]

slide-70
SLIDE 70

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Definition

Mean-field evolution

Independent (T = ∞) dynamics on ΩN induces on MN a continuous-time Markov chain (mN

t )t≥0 with generator

  • LNf
  • (m)

= 1 + m 2 N

  • f(m − 2N−1) − f(m)
  • +1 − m

2 N

  • f(m + 2N−1) − f(m)
  • This induces a dynamics on measures on MN

µN

t (f) = µN

etLN f

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SLIDE 71

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Definition

Mean-field evolution

Independent (T = ∞) dynamics on ΩN induces on MN a continuous-time Markov chain (mN

t )t≥0 with generator

  • LNf
  • (m)

= 1 + m 2 N

  • f(m − 2N−1) − f(m)
  • +1 − m

2 N

  • f(m + 2N−1) − f(m)
  • This induces a dynamics on measures on MN

µN

t (f) = µN

etLN f

slide-72
SLIDE 72

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Non-Gibbsianness

Mean-field single-site specification

(K¨ ulske and Le Ny)

Consider the single-spin conditional probabilities γN

t (σ1 | αN−1) := µN t (σ1 | σN−1) ,

with

◮ σ1 ∈ {−1, +1}, ◮ αN−1 ∈ MN−1, ◮ σN−1 ∈ ΩN−1 any configuration s.t. mN−1(σN−1) = αN−1

[By permutation invariance RHS independ of choice of σN−1]

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SLIDE 73

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Non-Gibbsianness

Gibbs and Non-Gibbs mean-field models

(K¨ ulske and Le Ny)

For fixed t ≥ 0: (a) A magnetization α ∈ [−1, 1] is good for µt if γt(· | α) := lim

N→∞ αN→ α

γN

t (· | αN−1),

◮ exists and is independent of the sequence αN →

α

◮ it is continuous in

α

for α in a neighbourhood of α (b) A magnetization α ∈ [−1, +1] is bad if it is not good (c) µt is Gibbs if it has no bad magnetizations

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SLIDE 74

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Non-Gibbsianness

Gibbs and Non-Gibbs mean-field models

(K¨ ulske and Le Ny)

For fixed t ≥ 0: (a) A magnetization α ∈ [−1, 1] is good for µt if γt(· | α) := lim

N→∞ αN→ α

γN

t (· | αN−1),

◮ exists and is independent of the sequence αN →

α

◮ it is continuous in

α

for α in a neighbourhood of α (b) A magnetization α ∈ [−1, +1] is bad if it is not good (c) µt is Gibbs if it has no bad magnetizations

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SLIDE 75

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Large deviations

Large-deviations: General definition

Informally: A family of measures (νN) satisfies a large-deviation principle if νN(A) ∼ e−N I(A)

◮ N is the LDP speed, I the rate function ◮ As a consequence, supp(νN) → argmin(I)

Formally: (νN) on a Borel space satisf. LDP with rate fcn I and speed N if lim inf

N→∞

1 N log νN(A) ≥ − inf

x∈A I(x)

for A open lim sup

N→∞

1 N log νN(A) ≤ − sup

x∈A

I(x) for A closed

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SLIDE 76

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Large deviations

Large-deviations: General definition

Informally: A family of measures (νN) satisfies a large-deviation principle if νN(A) ∼ e−N I(A)

◮ N is the LDP speed, I the rate function ◮ As a consequence, supp(νN) → argmin(I)

Formally: (νN) on a Borel space satisf. LDP with rate fcn I and speed N if lim inf

N→∞

1 N log νN(A) ≥ − inf

x∈A I(x)

for A open lim sup

N→∞

1 N log νN(A) ≤ − sup

x∈A

I(x) for A closed

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SLIDE 77

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Large deviations

Large-deviations: General definition

Informally: A family of measures (νN) satisfies a large-deviation principle if νN(A) ∼ e−N I(A)

◮ N is the LDP speed, I the rate function ◮ As a consequence, supp(νN) → argmin(I)

Formally: (νN) on a Borel space satisf. LDP with rate fcn I and speed N if lim inf

N→∞

1 N log νN(A) ≥ − inf

x∈A I(x)

for A open lim sup

N→∞

1 N log νN(A) ≤ − sup

x∈A

I(x) for A closed

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SLIDE 78

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Large deviations

LDP for mean-field Ising:

“Static” part: The family (µN) satisfies a LDP with speed N and rate IS − inf(IS) with IS(m) := H(m) + 1 + m 2 log(1 + m) + 1 − m 2 log(1 − m). Independent evolutions: Let P N = law of (mN

t )t≥0

Defined on the space of c` adl` ag trajectories; Skorohod topology

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SLIDE 79

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Large deviations

LDP for mean-field Ising:

“Static” part: The family (µN) satisfies a LDP with speed N and rate IS − inf(IS) with IS(m) := H(m) + 1 + m 2 log(1 + m) + 1 − m 2 log(1 − m). Independent evolutions: Let P N = law of (mN

t )t≥0

Defined on the space of c` adl` ag trajectories; Skorohod topology

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SLIDE 80

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Large deviations

LDP for mean-field evolutions

(Ermolaev and K¨ ulske)

(P N) restricted to [0, T] satisfies LDP with speed N and rate IT − inf(IT ) given by IT (φ) := IS(φ(0)) + IT

D(φ),

where IT

D(φ) :=

T

0 L(φ(s), ˙

φ(s)) ds if ˙ φ exists ∞

  • therwise

is the action integral with Lagrangian L(m, ˙ m) = −1 2

  • 4 (1 − m2) + ˙

m2 +1 2 ˙ m log

  • 4 (1 − m2) + ˙

m2 + ˙ m 2(1 − m)

  • + 1
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SLIDE 81

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Large deviations

LDP for mean-field evolutions

(Ermolaev and K¨ ulske)

(P N) restricted to [0, T] satisfies LDP with speed N and rate IT − inf(IT ) given by IT (φ) := IS(φ(0)) + IT

D(φ),

where IT

D(φ) :=

T

0 L(φ(s), ˙

φ(s)) ds if ˙ φ exists ∞

  • therwise

is the action integral with Lagrangian L(m, ˙ m) = −1 2

  • 4 (1 − m2) + ˙

m2 +1 2 ˙ m log

  • 4 (1 − m2) + ˙

m2 + ˙ m 2(1 − m)

  • + 1
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SLIDE 82

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Large deviations

LDP for conditioned mean-field evolutions

The family of measures on trajectory space QN

t,α( · ) := P N

(mN(s))0≤s≤t = ·

  • mN(t) = α
  • satisfies LDP with speed N and rate It,α − inf(It,α), with

It,α(mφ) = It(φ) if φt = α ∞

  • therwise

Hence, conditioned optimal trajectories correspond to argmin

φ: φ(t)=α

It(φ)

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SLIDE 83

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Large deviations

LDP for conditioned mean-field evolutions

The family of measures on trajectory space QN

t,α( · ) := P N

(mN(s))0≤s≤t = ·

  • mN(t) = α
  • satisfies LDP with speed N and rate It,α − inf(It,α), with

It,α(mφ) = It(φ) if φt = α ∞

  • therwise

Hence, conditioned optimal trajectories correspond to argmin

φ: φ(t)=α

It(φ)

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SLIDE 84

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results

The mean-field computational advantage

Simplifying feature:

◮ There is an explicit expression for

Ct,α(m) := inf

φ: φ(0)=m, φ(t)=α

It(φ)

◮ We have the identity

inf

m∈[−1,+1] Ct,α(m) =

inf

φ: φ(t)=α It(φ)

Hence, multiple conditioned trajectories ⇐ ⇒ multiple global minima of Ct,α(m)

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SLIDE 85

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results

The mean-field computational advantage

Simplifying feature:

◮ There is an explicit expression for

Ct,α(m) := inf

φ: φ(0)=m, φ(t)=α

It(φ)

◮ We have the identity

inf

m∈[−1,+1] Ct,α(m) =

inf

φ: φ(t)=α It(φ)

Hence, multiple conditioned trajectories ⇐ ⇒ multiple global minima of Ct,α(m)

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SLIDE 86

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results

  • I. Single optimal trajectory = Gibbsianness

α → γt(σ | α) is continuous at α0 if and only if

◮ It(φ) has a unique minimizing path

φ

◮ or, equivalently, Ct,α0(m) has a unique minimizing m.

Furthermore, in this case, the specification kernel equals γt(z | α) =

  • x∈{−1,+1} ex[J

φ(0)+h] pt(x, z)

  • x,y∈{−1,+1} ex[J

φ(0)+h] pt(x, y)

pt(·, ·) = kernel of Markov jump process on {−1, +1} with

◮ jumping rate 1 ◮ jump probabilities pt(i, ±i) = e−t cosh(t) sinh(t)

“If” part and for of γt proven by Ermolaev and K¨ ulske

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SLIDE 87

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results

  • I. Single optimal trajectory = Gibbsianness

α → γt(σ | α) is continuous at α0 if and only if

◮ It(φ) has a unique minimizing path

φ

◮ or, equivalently, Ct,α0(m) has a unique minimizing m.

Furthermore, in this case, the specification kernel equals γt(z | α) =

  • x∈{−1,+1} ex[J

φ(0)+h] pt(x, z)

  • x,y∈{−1,+1} ex[J

φ(0)+h] pt(x, y)

pt(·, ·) = kernel of Markov jump process on {−1, +1} with

◮ jumping rate 1 ◮ jump probabilities pt(i, ±i) = e−t cosh(t) sinh(t)

“If” part and for of γt proven by Ermolaev and K¨ ulske

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SLIDE 88

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results

  • I. Single optimal trajectory = Gibbsianness

α → γt(σ | α) is continuous at α0 if and only if

◮ It(φ) has a unique minimizing path

φ

◮ or, equivalently, Ct,α0(m) has a unique minimizing m.

Furthermore, in this case, the specification kernel equals γt(z | α) =

  • x∈{−1,+1} ex[J

φ(0)+h] pt(x, z)

  • x,y∈{−1,+1} ex[J

φ(0)+h] pt(x, y)

pt(·, ·) = kernel of Markov jump process on {−1, +1} with

◮ jumping rate 1 ◮ jump probabilities pt(i, ±i) = e−t cosh(t) sinh(t)

“If” part and for of γt proven by Ermolaev and K¨ ulske

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SLIDE 89

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results

  • II. Short-term Gibbsianness

Theorem If J ≤ 1 the evolved measures µt are Gibbs for all t ≥ 0

◮ Proven by K¨

ulske and Le Ny and K¨ ulske and Ermolaev

◮ Note that 1 = βMF cr

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SLIDE 90

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results

  • II. Short-term Gibbsianness

Theorem If J ≤ 1 the evolved measures µt are Gibbs for all t ≥ 0

◮ Proven by K¨

ulske and Le Ny and K¨ ulske and Ermolaev

◮ Note that 1 = βMF cr

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SLIDE 91

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results

  • III. Case J > 1, h = 0

Consider the critical time Ψc(J) :=

  • 1

2 acoth(2J − 1)

if 1 < J ≤ 3

2,

t∗(J) implicitly calculable if J > 3

2,

Then:

◮ t < Ψc: Evolved measure µt is Gibbs ◮ t > Ψc: Discontinuity at α = 0; two optimal trajectories ±

φ

◮ If Λt,0(J) = cone between the trajectories ±ˆ

φ

◮ No trajectory can penetrate Λt,0(J) ◮ For J ≤ 3/2 the map t → Λt,0(J) is continuous ◮ For J > 3/2 the map t → Λt,0(J) is continuous except at

t = Ψc where it exhibits a right-continuous jump

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SLIDE 92

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results

  • III. Case J > 1, h = 0

Consider the critical time Ψc(J) :=

  • 1

2 acoth(2J − 1)

if 1 < J ≤ 3

2,

t∗(J) implicitly calculable if J > 3

2,

Then:

◮ t < Ψc: Evolved measure µt is Gibbs ◮ t > Ψc: Discontinuity at α = 0; two optimal trajectories ±

φ

◮ If Λt,0(J) = cone between the trajectories ±ˆ

φ

◮ No trajectory can penetrate Λt,0(J) ◮ For J ≤ 3/2 the map t → Λt,0(J) is continuous ◮ For J > 3/2 the map t → Λt,0(J) is continuous except at

t = Ψc where it exhibits a right-continuous jump

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SLIDE 93

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results

  • III. Case J > 1, h = 0

Consider the critical time Ψc(J) :=

  • 1

2 acoth(2J − 1)

if 1 < J ≤ 3

2,

t∗(J) implicitly calculable if J > 3

2,

Then:

◮ t < Ψc: Evolved measure µt is Gibbs ◮ t > Ψc: Discontinuity at α = 0; two optimal trajectories ±

φ

◮ If Λt,0(J) = cone between the trajectories ±ˆ

φ

◮ No trajectory can penetrate Λt,0(J) ◮ For J ≤ 3/2 the map t → Λt,0(J) is continuous ◮ For J > 3/2 the map t → Λt,0(J) is continuous except at

t = Ψc where it exhibits a right-continuous jump

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SLIDE 94

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results

  • III. Case J > 1, h = 0

Consider the critical time Ψc(J) :=

  • 1

2 acoth(2J − 1)

if 1 < J ≤ 3

2,

t∗(J) implicitly calculable if J > 3

2,

Then:

◮ t < Ψc: Evolved measure µt is Gibbs ◮ t > Ψc: Discontinuity at α = 0; two optimal trajectories ±

φ

◮ If Λt,0(J) = cone between the trajectories ±ˆ

φ

◮ No trajectory can penetrate Λt,0(J) ◮ For J ≤ 3/2 the map t → Λt,0(J) is continuous ◮ For J > 3/2 the map t → Λt,0(J) is continuous except at

t = Ψc where it exhibits a right-continuous jump

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SLIDE 95

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results

  • III. Case J > 1, h = 0

Consider the critical time Ψc(J) :=

  • 1

2 acoth(2J − 1)

if 1 < J ≤ 3

2,

t∗(J) implicitly calculable if J > 3

2,

Then:

◮ t < Ψc: Evolved measure µt is Gibbs ◮ t > Ψc: Discontinuity at α = 0; two optimal trajectories ±

φ

◮ If Λt,0(J) = cone between the trajectories ±ˆ

φ

◮ No trajectory can penetrate Λt,0(J) ◮ For J ≤ 3/2 the map t → Λt,0(J) is continuous ◮ For J > 3/2 the map t → Λt,0(J) is continuous except at

t = Ψc where it exhibits a right-continuous jump

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SLIDE 96

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results

Graphic summary: h = 0, α = 0

t1

  • c

m(t,) ^ t m(t,) ^ t

  • c

m*

  • m*

Forbidden Region

m(t,) ^ t

  • c

t2 m

  • m

Ct,0 m Ct,0 m m*

  • m*

Ct,0 m

t < Ψc t = Ψc t > Ψc First row: Minimizing trajectories for (J, h) = (1.6, 0) Second row: Corresponding plots of m → Ct,0(m)

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SLIDE 97

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results

  • IV. Bad magnetizations as function of time
  • c

J=1.4, h=0

t

J=2.5, h=0

t

  • c

tB

  • U

UB

  • UB

J=1.4, h=0.29

t

*

  • U

U

B

tB

J=2.5, h=0.1

t

  • U

* U

B

MT L

B

tB sB

MB T L

1 < J ≤ 3

2

J > 3

2

SB = trifurcation point, rest of the line = bifurcation

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SLIDE 98

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Kac models: Definition

Kac models: Basic definitions

◮ ∆d n := Zd/nZd = the discrete torus of size n ◮ Ωn := {−1, +1}∆d

n = Ising-spin configurations on ∆d

n ◮ Kac-type Hamiltonian:

Hn(σ) := − 1

2nd

  • x,y∈∆d

n

J x−y

n

  • σ(x)σ(y) −
  • x∈∆d

n

h( x

n) σ(x)

[J ≥ 0 symmetric]

◮ Gibbs measure associated with Hn:

µn(dσ) := e−βHn(σ) Zn dσ

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SLIDE 99

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Kac models: Definition

Kac models: Basic definitions

◮ ∆d n := Zd/nZd = the discrete torus of size n ◮ Ωn := {−1, +1}∆d

n = Ising-spin configurations on ∆d

n ◮ Kac-type Hamiltonian:

Hn(σ) := − 1

2nd

  • x,y∈∆d

n

J x−y

n

  • σ(x)σ(y) −
  • x∈∆d

n

h( x

n) σ(x)

[J ≥ 0 symmetric]

◮ Gibbs measure associated with Hn:

µn(dσ) := e−βHn(σ) Zn dσ

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SLIDE 100

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Kac models: Definition

Continuum limit

◮ Td := Rd/Zd, the d-dimensional unit torus ◮ Td n := 1 n∆d n = (1/n)-discretization of Td ◮ M(Td n) [M(Td)] = signed measures on Td n [Td] (TV ≤ 1)

The empirical density of σ ∈ Ωn inside Λ ⊆ ∆d

n is

πn

Λ : Ωn → M(Td n) ⊆ M(Td)

πn

Λ(σ) := 1

|Λ|

  • x∈Λ

σ(x)δx/n

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SLIDE 101

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Kac models: Definition

Continuum limit

◮ Td := Rd/Zd, the d-dimensional unit torus ◮ Td n := 1 n∆d n = (1/n)-discretization of Td ◮ M(Td n) [M(Td)] = signed measures on Td n [Td] (TV ≤ 1)

The empirical density of σ ∈ Ωn inside Λ ⊆ ∆d

n is

πn

Λ : Ωn → M(Td n) ⊆ M(Td)

πn

Λ(σ) := 1

|Λ|

  • x∈Λ

σ(x)δx/n

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SLIDE 102

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Kac models: Definition

Induced objects. Profiles

Via πn we define induced Gibbs measures on M(Td

n):

ˇ µn = µn ◦ (πn)−1 and rewrite Hn(σ) = ndH(πn(σ)) with H(ν) = −

  • 1

2J ∗ ν + h, ν

  • A measure on M(Td) of the form α λ with

◮ λ Lebesgue measure ◮ α ∈ B density function, with B=unit ball in L∞(Td)

will be referred as a profile α ∈ B

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SLIDE 103

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Kac models: Definition

Induced objects. Profiles

Via πn we define induced Gibbs measures on M(Td

n):

ˇ µn = µn ◦ (πn)−1 and rewrite Hn(σ) = ndH(πn(σ)) with H(ν) = −

  • 1

2J ∗ ν + h, ν

  • A measure on M(Td) of the form α λ with

◮ λ Lebesgue measure ◮ α ∈ B density function, with B=unit ball in L∞(Td)

will be referred as a profile α ∈ B

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SLIDE 104

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Kac models: Definition

Single-site Kac specifications

Given

◮ A (continuum) site u ∈ Td ◮ A probability measure ρn on Ωn ◮ A measure αu n−1 ∈ M(Td n \ ⌊nu⌋)

The single-site conditional probability at site ⌊nu⌋ ∈ Td

n is

γu,n ·

  • αu

n−1

  • := ρn

σ(⌊nu⌋) = ·

  • πu,n(σ) = αu

n−1

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SLIDE 105

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Kac models: Definition

Single-site Kac specifications

Given

◮ A (continuum) site u ∈ Td ◮ A probability measure ρn on Ωn ◮ A measure αu n−1 ∈ M(Td n \ ⌊nu⌋)

The single-site conditional probability at site ⌊nu⌋ ∈ Td

n is

γu,n ·

  • αu

n−1

  • := ρn

σ(⌊nu⌋) = ·

  • πu,n(σ) = αu

n−1

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SLIDE 106

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Kac models: Definition

Gibbs and no-Gibbs Kac measures

(a) A profile α ∈ B is good for (ρn) γu ·

  • α
  • := lim

n→∞ γu,n

·

  • αu

n−1

  • ◮ exists and is independent of the sequence αu

n−1 →

αλ

◮ it is continuous in

α

for α in a neighbourhood of α (b) A profile α ∈ B is bad if it is not good (c) (ρn) is Gibbs if it has no bad profiles

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SLIDE 107

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Kac models: Definition

Gibbs and no-Gibbs Kac measures

(a) A profile α ∈ B is good for (ρn) γu ·

  • α
  • := lim

n→∞ γu,n

·

  • αu

n−1

  • ◮ exists and is independent of the sequence αu

n−1 →

αλ

◮ it is continuous in

α

for α in a neighbourhood of α (b) A profile α ∈ B is bad if it is not good (c) (ρn) is Gibbs if it has no bad profiles

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SLIDE 108

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Kac models: Definition

LDP for (“static”) Kac measures

(Comets)

(ˇ µn) satisfies an LDP with

◮ speed nd ◮ rate function IS − infM(Td) IS with

IS(ν) :=

  • −β

1

2J ∗ α + h, αλ

  • + Φ ◦ α, λ

if ν = αλ, α ∈ B ∞

  • therwise

where Φ is the relative entropy Φ(m) := 1+m

2

log(1 + m) + 1−m

2

log(1 − m), m ∈ [−1, +1].

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SLIDE 109

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Kac models: Definition

LDP for conditioned Kac evolutions

Let P n := law of (πn

s )s≥0 conditional on πn 0 ∼ ˇ

µn, and Qn

t,α(·) := P n

(πn

s )s∈[0,t] ∈ · | πn t = αn

  • ,

with αn ∈ Mn the element closest to αλ. Then For t ≥ 0 and α ∈ B, (Qn

t,α′)n∈N satisfies an LDP with ◮ speed nd ◮ rate function It,α − infD[0,t](M(Td)) It,α with

It,α(φ) := IS(φ0) + It

D(φ)

if φt ≡ α ∞,

  • therwise.

with It

D(φ) given by the integral of an explicit Lagrangian

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SLIDE 110

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Kac models: Definition

LDP for conditioned Kac evolutions

Let P n := law of (πn

s )s≥0 conditional on πn 0 ∼ ˇ

µn, and Qn

t,α(·) := P n

(πn

s )s∈[0,t] ∈ · | πn t = αn

  • ,

with αn ∈ Mn the element closest to αλ. Then For t ≥ 0 and α ∈ B, (Qn

t,α′)n∈N satisfies an LDP with ◮ speed nd ◮ rate function It,α − infD[0,t](M(Td)) It,α with

It,α(φ) := IS(φ0) + It

D(φ)

if φt ≡ α ∞,

  • therwise.

with It

D(φ) given by the integral of an explicit Lagrangian

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SLIDE 111

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Kac models: Results

Results for Kac models

(A) For general Glauber dynamics: Gibbsianness ⇐ ⇒ unique minimizing path (B) For independent spin flips: Let J :=

  • Td J(u)du, then

(i) Short-time Gibbs: ∃ t0 = t0(J, h) s.t. no bifurcation in [0, t0] (ii) Mean-Field behaviour: If h ≡ c ∈ [0, ∞) and α′ ≡ c′ ∈ [−1, +1] then bifurcation ∼ MF with (JMF, hMF) = (βJ, βc); α = c′

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SLIDE 112

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Kac models: Results

Results for Kac models

(A) For general Glauber dynamics: Gibbsianness ⇐ ⇒ unique minimizing path (B) For independent spin flips: Let J :=

  • Td J(u)du, then

(i) Short-time Gibbs: ∃ t0 = t0(J, h) s.t. no bifurcation in [0, t0] (ii) Mean-Field behaviour: If h ≡ c ∈ [0, ∞) and α′ ≡ c′ ∈ [−1, +1] then bifurcation ∼ MF with (JMF, hMF) = (βJ, βc); α = c′

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SLIDE 113

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Kac models: Results

Results for Kac models

(A) For general Glauber dynamics: Gibbsianness ⇐ ⇒ unique minimizing path (B) For independent spin flips: Let J :=

  • Td J(u)du, then

(i) Short-time Gibbs: ∃ t0 = t0(J, h) s.t. no bifurcation in [0, t0] (ii) Mean-Field behaviour: If h ≡ c ∈ [0, ∞) and α′ ≡ c′ ∈ [−1, +1] then bifurcation ∼ MF with (JMF, hMF) = (βJ, βc); α = c′

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SLIDE 114

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End

Conclusions

◮ New paradigm seems to work ◮ However: needs LDP in spaces of trajectories of measure ◮ Practical consequences (numerics, other phenomena)?