Weierstrass Institute forApplied Analysis and Stochastics

Dynamical Gibbs-non-Gibbs transitions for a continuous

spin model

Benedikt Jahnel & Christof Külske

The Widom-Rowlinson model (WRM) under independent spin flip

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General setting: Gibbs Point Processes

Local state space: E = 1, . . . , q Space of particles:

Ω = ω ⊂ Rd : |ωΛ| = #ω ∩ Λ <∞ for all bounded Λ ⊂ Rd Space of colored particles: Ω = ω = (ω(1), . . . , ω(q)) : ω(i) ∈ Ω Grey configuration ω of a colored configuration ω: ω = ω(1) ∪ · · · ∪ ω(q)

Color of the particle x ∈ Rd: σx ∈ E Colored particle configuration ωΛ correspond to (ωΛ, σωΛ)

σ-algebra generated by the counting variables Ω 3 ω 7→ #(ω ∩ Λ) for bounded and

measurable Λ b Rd: F on Ω

σ-algebras in finite-volume: FΛ on ΩΛ

σ-algebras for colored particles: F ,FΛ

Observables: f ∈ FbΛ if f measurable w.r.t. FΛ and bounded w.r.t. the supremum norm

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Gibbsian setup: Specifications and τ -topology

Specification:

Family of proper probability kernels γ = (γΛ)ΛbRd obeying consistency

γ∆(γΛ(dω|·)|ω) = γ∆(dω|ω)

for all measurable volumes Λ ⊂ ∆ b Rd and ω ∈ Ω.

Candidate system for conditional probabilities of infinite-volume Gibbs measures µ

to be defined by DLR equations

µ(γΛ(f |·)) = µ(f), f ∈ Fb.

Measurability: γΛ(f |·) ∈ FΛc

τ -topology: ω′ ⇒ ω, iff f(ω′)→ f(ω) for all local f ∈ Fb.

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Poisson point processes

Homogenous Poisson point process: random point cloud ω ∈ Ω with:

1. Point clouds in disjoint areas are stochastically independent

2. Number of points in Λ ⊂ Rd is Poisson distributed with parameter λ|Λ|:

P (ω has k points in Λ) = e−λ|Λ|(λ|Λ|)k

k!

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Gibbsian setup: Potentials

Specifications as Poisson modification given via Boltzmann weights

γΛ(dωΛ|ωΛc) := PΛ(dωΛ)e−HΛ(ωΛωΛc )Z−1Λ (ωΛc)

where Hamiltonian HΛ given via interaction potential Φ,

HΛ(ω) =∑

ηbω: η∩Λ 6=∅

Φ(η,ω)

Potts gas: Φ(η,ω) = δη=x,y[δσx 6=σyϕ(x− y) + ψ(x− y)]

problem:∑ηbω: η∩Λ6=∅ Φ(η,ω) maybe not well-defined

need to consider admissible boundary conditions Ω∗

measurability and summability properties of Φ?

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Example of Gibbs point process: WRM

Spatial dimension: d ≥ 2

Two-color spin space: E = −,+ Two-color homogenous Poisson point process, with intensities λ+ for plus colors and λ−

for minus colors: base measure P

Color constraint: discs of radius a between different colors are forbidden to overlap

Widom-Rowlinson specification (hard core & strictly local)

γΛ(dωΛ|ωΛc) := PΛ(dωΛ)χ(ωΛωΛc)Z−1Λ (ωΛc).

Where indicator χ is either one (or zero) if interspecies distance is bigger or equal than 2a for

all particles (or not).

B. Widom and J. S. Rowlinson ’70: New model for the study of liquid–vapor phase transitions, The

Journal of Chemical Physics 52.

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Specifications for the WRM

Alternative description:

Sample colorless (grey) points according to Poisson point process P with intensity

λ+ + λ−.

Sample colors according to independent Bernoulli measures U on the color-space E,

with probability to see color + given by λ+/(λ+ + λ−).

γΛ(dωΛ|ωΛc) = PΛ(dωΛ)U(dσωΛ)χ(ωΛωΛc)Z−1Λ (ωΛc)

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Phase transition for the WRM

In spatial dimensions d ≥ 2 the WRM exhibits a phase-transition in symmetric

high-intensity regime.

FKG implies existence of limit limΛ↑Rd γΛ(dωΛ|±Λc) = µ±(dω).

Peierls argument: D. Ruelle ’71: Existence of a Phase Transition in a Continuous Classical System,

Phys. Rev. Lett. 27.

Random cluster representation: J. T. Chayes, L. Chayes and R. Kotecký ’95: The analysis of the

Widom-Rowlinson model by stochastic geometric methods, Comm. Math. Phys. 172.

Random cluster representation: G. Giacomin, J. L. Lebowitz and C. Maes ’95: Agreement

percolation and phase coexistence in some Gibbs systems, J. Statist. Phys. 80

Existence of infinite-range Potts gas: H.-O. Georgii and O. Häggström ’96: Phase transition in

continuum Potts models, Comm. Math. Phys. 181.

Lattice WRM: Y. Higuchi and M. Takei ’04: Some results on the phase structure of the

two-dimensional Widom-Rowlinson model , Osaka J. Math. 41.

General existence theory: D. Dereudre, R. Drouilhet and H.-O. Georgii ’12: Existence of Gibbsian

point processes with geometry-dependent interactions, Probab. Theory Related Fields 153.

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The dynamics: Symmetric independent spin flips

Keeping the locations fixed, we apply independently over the points the transition kernel

pt(+,+) =1

2(1 + e−2t).

Realization of the WRM in the phase transition regime under independent spin-flip at time zero

(left) and for some positive time (right).

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Continuation of GnG program

Questions:

For which λ+, λ−, t has µt a local specification?

What is the measure of bad points for an optimal choice of a specification?

Selected results for GnG under spin flip:

Lattice: A. C. D. van Enter, R. Fernández, F. den Hollander and F. Redig ’02: Possible loss & re-

covery of Gibbsianness during the stochastic evolution of Gibbs measures, Comm. Math. Phys. 226.

Mean field: V. Ermolaev and C. Külske ’10: Low-temperature dynamics of the Curie-Weiss model:

periodic orbits, multiple histories, and loss of Gibbsianness, J. Stat. Phys. 141.

Trees: A. C. D. van Enter, V. N. Ermolaev, G. Iacobelli and C. Külske ’12: Gibbs-non-Gibbs

properties for evolving Ising models on trees, Ann. Inst. Henri Poincaré Probab. Stat. 48.

Kac: R. Fernández, F. den Hollander and J. Martínez ’14: Variational description of Gibbs-non-Gibbs

dynamical transitions for spin-flip systems with a Kac-type interaction, J. Stat. Phys. 156.

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Continuation of GnG program

Questions:

For which λ+, λ−, t has µt a local specification?

What is the measure of bad points for an optimal choice of a specification?

Selected results for GnG under spin flip:

Lattice: A. C. D. van Enter, R. Fernández, F. den Hollander and F. Redig ’02: Possible loss & re-

covery of Gibbsianness during the stochastic evolution of Gibbs measures, Comm. Math. Phys. 226.

Mean field: V. Ermolaev and C. Külske ’10: Low-temperature dynamics of the Curie-Weiss model:

periodic orbits, multiple histories, and loss of Gibbsianness, J. Stat. Phys. 141.

Trees: A. C. D. van Enter, V. N. Ermolaev, G. Iacobelli and C. Külske ’12: Gibbs-non-Gibbs

properties for evolving Ising models on trees, Ann. Inst. Henri Poincaré Probab. Stat. 48.

Kac: R. Fernández, F. den Hollander and J. Martínez ’14: Variational description of Gibbs-non-Gibbs

dynamical transitions for spin-flip systems with a Kac-type interaction, J. Stat. Phys. 156.

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Definitions: Good & bad points, quasilocality, almost-sure quasilocality

Definition

Let γ be a specification. A configuration ω ∈ Ω is called good for γ iff for any x ∈ Rd and

0 < r <∞ and any observable f ∈ FbBr(x) we have∣∣γBr(x)(f |ω′Br(x)c)− γBr(x)(f |ωBr(x)c)

∣∣→ 0

as ω′ ⇒ ω.

Set of good configurations: Ω(γ)

Set of bad configurations: Ω \Ω(γ)

γ is called quasilocal iff Ω(γ) = Ω

µ is called quasilocally Gibbs (q) iff there exists quasilocal γ for µ

µ is called a.s.-quasilocally Gibbs (asq) iff there exists γ for µ with µ(Ω(γ)) = 1

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Overview of results: Immediate loss and sharp recovery

Reentrance time into Gibbs tG := 12

logλ++λ−λ+−λ−

for λ+ > λ−

High-intensity (percolating) regime: µ(Br(x)↔∞) > 0 for some ball Br(x)

Low-intensity (non-percolating) regime: µ(Br(x)↔∞) = 0 for all Br(x)

Table: Gibbsian transitions in time and intensity.

G(γ) time high intensity low intensity

0 < t < tG non-asq asq, non-q

λ+ > λ− µ+ t = tG asq, non-q asq, non-q

tG < t ≤ ∞ q q

λ+ = λ−µ 0 < t <∞ non-asq asq, non-q

µ+ t =∞ non-asq asq, non-q

BJ and Christof Külske ’17: The Widom-Rowlinson model under spin flip: Immediate loss and sharp

recovery of quasilocality , Annals of Applied Probability, Vol. 27, No. 6, 3845-3892

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Take home message

tG time

intensity

a.s. quasilocal

non-a.s. quasilocalquasilocal

Asymmetric model

time

intensity

a.s. quasilocal

non-a.s. quasilocal

Symmetric model

Illustration of Gibbs-non-Gibbs transitions in time and intensity for the WRM under independent

spin flip.

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Ideas of Proof: Finite-volume conditional probabilities

Time-evolved WRM in Λ ⊂ Rd with (not-time evolved) boundary condition ωΛc

µωΛc

t,Λ (f) =

∫γΛ(dωΛ|ωΛc)

∫pt(σωΛ , dσωΛ)f(ωΛ).

Derive and analyze a cluster-representation for conditional measures

µωΛc

t,Λ (f |ωΛ\B)

in a window B ⊂ Rd conditional to the outside Λ \B for some observable f ∈ FbB .

Observation: Discontinuities may arise only on large clusters from

color perturbations or spatial perturbations.

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Ideas of Proof: Finite-volume conditional probabilities

Time-evolved WRM in Λ ⊂ Rd with (not-time evolved) boundary condition ωΛc

µωΛc

t,Λ (f) =

∫γΛ(dωΛ|ωΛc)

∫pt(σωΛ , dσωΛ)f(ωΛ).

Derive and analyze a cluster-representation for conditional measures

µωΛc

t,Λ (f |ωΛ\B)

in a window B ⊂ Rd conditional to the outside Λ \B for some observable f ∈ FbB .

Observation: Discontinuities may arise only on large clusters from

color perturbations or spatial perturbations.

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Outlook

DAAD project on Gibbs measures on random point processes (07/2018 - 07/2022)

Jointly with Prof. Dr. Wolfgang König, Dr. Alex Opoku, Dr. Kwabena Doku-Amponsah

Immediate loss of Gibbsianness = effect of the hard-core repulsion?

Softer repulsion = short-time Gibbsianness?

C. Külske and A.A. Opoku ’08: The Posterior metric and the Goodness of Gibbsianness for

transforms of Gibbs measures, Elec. Journ. Prob. 13.

Dobrushin uniqueness / strong field condition for Gibbs point processes?

Translation invariant potential representations for Gibbsian specifications in the

continuum?

Absence of Gibbsian specifications = failure of variational principle?

Attractor properties for Glauber dynamics for Gibbs point processes?

. . .

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Thank you.

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