Dynamical GibbsnonGibbs transitions for a continuous ... · percolation and phase coexistence in...
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Weierstrass Institute forApplied Analysis and Stochastics
Dynamical GibbsnonGibbs transitions for a continuous
spin model
Benedikt Jahnel & Christof Külske

The WidomRowlinson 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∆(dωω̃)
for all measurable volumes Λ ⊂ ∆ b Rd and ω̃ ∈ Ω.� Candidate system for conditional probabilities of infinitevolume 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 λΛ:
P ({ω has k points in Λ}) = e−λΛ (λΛ)k
k!
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Gibbsian setup: Potentials
� Specifications as Poisson modification given via Boltzmann weights
γΛ(dωΛωΛc) := 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 welldefined
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� Twocolor spin space: E = {−,+}� Twocolor 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
WidomRowlinson specification (hard core & strictly local)
γΛ(dωΛωΛc) := 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 colorspace E,
with probability to see color + given by λ+/(λ+ + λ−).
γΛ(dωΛωΛc) = 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 phasetransition in symmetrichighintensity 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
WidomRowlinson 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 infiniterange 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
twodimensional WidomRowlinson model , Osaka J. Math. 41.
� General existence theory: D. Dereudre, R. Drouilhet and H.O. Georgii ’12: Existence of Gibbsian
point processes with geometrydependent 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 spinflip 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: Lowtemperature dynamics of the CurieWeiss 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: GibbsnonGibbs
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 GibbsnonGibbs
dynamical transitions for spinflip systems with a Kactype 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: Lowtemperature dynamics of the CurieWeiss 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: GibbsnonGibbs
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 GibbsnonGibbs
dynamical transitions for spinflip systems with a Kactype interaction, J. Stat. Phys. 156.
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Definitions: Good & bad points, quasilocality, almostsure 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−
� Highintensity (percolating) regime: µ(Br(x)↔∞) > 0 for some ball Br(x)� Lowintensity (nonpercolating) 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 nonasq asq, nonq
λ+ > λ− µ+ t = tG asq, nonq asq, nonq
tG < t ≤ ∞ q q
λ+ = λ−µ 0 < t

Take home message
tG time
intensity
a.s. quasilocal
nona.s. quasilocalquasilocal
Asymmetric model
time
intensity
a.s. quasilocal
nona.s. quasilocal
Symmetric model
Illustration of GibbsnonGibbs transitions in time and intensity for the WRM under independent
spin flip.
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Ideas of Proof: Finitevolume conditional probabilities
� Timeevolved WRM in Λ ⊂ Rd with (nottime evolved) boundary condition ωΛc
µωΛct,Λ (f) =
∫γΛ(dωΛωΛc)
∫pt(σωΛ , dσ̂ωΛ)f(ω̂Λ).
� Derive and analyze a clusterrepresentation 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: Finitevolume conditional probabilities
� Timeevolved WRM in Λ ⊂ Rd with (nottime evolved) boundary condition ωΛc
µωΛct,Λ (f) =
∫γΛ(dωΛωΛc)
∫pt(σωΛ , dσ̂ωΛ)f(ω̂Λ).
� Derive and analyze a clusterrepresentation 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 DokuAmponsah
� Immediate loss of Gibbsianness = effect of the hardcore repulsion?
� Softer repulsion = shorttime 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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