File size: 2,623 Bytes
f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 | import OAI.Probability.FreeIsing.Estimates.CenteredWordAction
import OAI.Probability.FreeIsing.Factors.Observable
namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367.FactorObservable
variable {d : ℕ}
variable {Phi : (TreeVertex d → ℝ) → (TreeVertex d → IsingSpin)}
lemma wordOperator_spinLp_root (hPhi : Measurable Phi)
(heq : ∀ g : TreeAut d,
∀ᵐ U ∂labelMeasure d, Phi (relabel g U) = relabel g (Phi U))
(v : TreeVertex d) :
RadialWalk.wordOperator TreeKoopman.generatorOperator v
(spinLp hPhi (ConnectedHull.root d)) = spinLp hPhi v :=
TreeKoopman.wordOperator_toLp_root (field Phi) (equivariant_field heq)
(fun v => memLp_field hPhi v 2) v
lemma wordOperator_centeredSpinLp_root {theta : ℝ} (hPhi : Measurable Phi)
(hlaw : HasFreeIsingLaw theta (Measure.map Phi (labelMeasure d)))
(heq : ∀ g : TreeAut d,
∀ᵐ U ∂labelMeasure d, Phi (relabel g U) = relabel g (Phi U))
(v : TreeVertex d) :
RadialWalk.wordOperator TreeKoopman.centeredGenerator v
(centeredSpinLp hPhi hlaw (ConnectedHull.root d)) = centeredSpinLp hPhi hlaw v := by
apply Subtype.ext
rw [TreeKoopman.centered_wordOperator_apply_coe]
exact wordOperator_spinLp_root hPhi heq v
lemma inner_centeredWordOperator_root {theta : ℝ} (hPhi : Measurable Phi)
(hlaw : HasFreeIsingLaw theta (Measure.map Phi (labelMeasure d)))
(heq : ∀ g : TreeAut d,
∀ᵐ U ∂labelMeasure d, Phi (relabel g U) = relabel g (Phi U))
(hlo : -1 ≤ theta) (hhi : theta ≤ 1) (v : TreeVertex d) :
inner ℝ (centeredSpinLp hPhi hlaw (ConnectedHull.root d))
(RadialWalk.wordOperator TreeKoopman.centeredGenerator v
(centeredSpinLp hPhi hlaw (ConnectedHull.root d))) = theta ^ v.1.length := by
rw [wordOperator_centeredSpinLp_root hPhi hlaw heq]
exact inner_centeredSpinLp_root hPhi hlaw hlo hhi v
end Problem367.FactorObservable
namespace Problem367
theorem IsFactorOfIID.exists_centered_word_observable {d : ℕ} {theta : ℝ}
(hfactor : IsFactorOfIID d theta) (hlo : -1 ≤ theta) (hhi : theta ≤ 1) :
∃ f : centeredL2 (labelMeasure d), ‖f‖ = 1 ∧
∀ v : TreeVertex d,
inner ℝ f (RadialWalk.wordOperator TreeKoopman.centeredGenerator v f) =
theta ^ v.1.length := by
obtain ⟨Phi, hPhi, hlaw, heq⟩ := hfactor
refine ⟨FactorObservable.centeredSpinLp hPhi hlaw (ConnectedHull.root d),
FactorObservable.norm_centeredSpinLp hPhi hlaw _, ?_⟩
intro v
exact FactorObservable.inner_centeredWordOperator_root hPhi hlaw heq hlo hhi v
end Problem367
end
end OAI
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