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

WRM under spin flip · March 21st, 2018 · Page 2

General setting: Gibbs Point Processes

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

Ω = {ω ⊂ Rd : |ωΛ| = #{ω ∩ Λ}

Gibbsian setup: Specifications and τ -topology

� Specification:

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

γ∆(γΛ(dω|·)|ω̃) = γ∆(dω|ω̃)

for all measurable volumes Λ ⊂ ∆ b Rd and ω̃ ∈ 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 symmetrichigh-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.

WRM under spin flip · March 21st, 2018 · Page 11

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 and0 < r

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 intensity0 < 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

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

µωΛct,Λ (f) =

∫γΛ(dωΛ|ωΛc)

∫pt(σωΛ , dσ̂ωΛ)f(ω̂Λ).

� Derive and analyze a cluster-representation for conditional measures

µωΛct,Λ (f |ω̂Λ\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

µωΛct,Λ (f) =

∫γΛ(dωΛ|ωΛc)

∫pt(σωΛ , dσ̂ωΛ)f(ω̂Λ).

� Derive and analyze a cluster-representation for conditional measures

µωΛct,Λ (f |ω̂Λ\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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