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Weierstrass Institute for Applied Analysis and Stochastics Dynamical Gibbs-non-Gibbs transitions for a continuous spin model Benedikt Jahnel & Christof Külske

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  • 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.

    WRM under spin flip · March 21st, 2018 · Page 4

  • 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!

    WRM under spin flip · March 21st, 2018 · Page 5

  • 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?

    WRM under spin flip · March 21st, 2018 · Page 6

  • 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.

    WRM under spin flip · March 21st, 2018 · Page 7

  • 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)

    WRM under spin flip · March 21st, 2018 · Page 8

  • 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.

    WRM under spin flip · March 21st, 2018 · Page 9

  • 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).

    WRM under spin flip · March 21st, 2018 · Page 10

  • 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

  • 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.

    WRM under spin flip · March 21st, 2018 · Page 14

  • 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.

    WRM under spin flip · March 21st, 2018 · Page 15

  • 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.

    WRM under spin flip · March 21st, 2018 · Page 15

  • 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?

    � . . .

    WRM under spin flip · March 21st, 2018 · Page 16

  • Thank you.

    WRM under spin flip · March 21st, 2018 · Page 17

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