| import OAI.Probability.FreeIsing.Spins.Moments |
| import OAI.Probability.FreeIsing.Trees.Koopman |
| import OAI.Probability.FreeIsing.Trees.RootCorrelation |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory |
|
|
| namespace Problem367.FactorObservable |
|
|
| variable {d : ℕ} |
| variable {Phi : (TreeVertex d → ℝ) → (TreeVertex d → IsingSpin)} |
|
|
| def field (Phi : (TreeVertex d → ℝ) → (TreeVertex d → IsingSpin)) |
| (U : TreeVertex d → ℝ) (v : TreeVertex d) : ℝ := |
| (Phi U v).toReal |
|
|
| lemma measurable_field (hPhi : Measurable Phi) (v : TreeVertex d) : |
| Measurable (fun U => field Phi U v) := |
| measurable_spinToReal.comp ((measurable_pi_apply v).comp hPhi) |
|
|
| lemma equivariant_field |
| (heq : ∀ g : TreeAut d, |
| ∀ᵐ U ∂labelMeasure d, Phi (relabel g U) = relabel g (Phi U)) : |
| ∀ g : TreeAut d, |
| ∀ᵐ U ∂labelMeasure d, field Phi (relabel g U) = relabel g (field Phi U) := by |
| intro g |
| filter_upwards [heq g] with U hU |
| funext v |
| exact congrArg IsingSpin.toReal (congrFun hU v) |
|
|
| lemma memLp_field (hPhi : Measurable Phi) (v : TreeVertex d) (p : ENNReal) : |
| MemLp (fun U => field Phi U v) p (labelMeasure d) := by |
| apply MemLp.of_bound (measurable_field hPhi v).aestronglyMeasurable 1 |
| exact Filter.Eventually.of_forall (fun U => (norm_spinToReal (Phi U v)).le) |
|
|
| def spinLp (hPhi : Measurable Phi) (v : TreeVertex d) : Lp ℝ 2 (labelMeasure d) := |
| (memLp_field hPhi v 2).toLp (fun U => field Phi U v) |
|
|
| lemma coe_spinLp (hPhi : Measurable Phi) (v : TreeVertex d) : |
| (spinLp hPhi v : (TreeVertex d → ℝ) → ℝ) =ᵐ[labelMeasure d] |
| (fun U => field Phi U v) := |
| (memLp_field hPhi v 2).coeFn_toLp |
|
|
| lemma norm_spinLp (hPhi : Measurable Phi) (v : TreeVertex d) : |
| ‖spinLp hPhi v‖ = 1 := by |
| rw [spinLp, Lp.norm_toLp] |
| have heq : eLpNorm (fun U => field Phi U v) 2 (labelMeasure d) = |
| eLpNorm (fun _ : TreeVertex d → ℝ => (1 : ℝ)) 2 (labelMeasure d) := by |
| apply eLpNorm_congr_norm_ae (measurable_field hPhi v).aestronglyMeasurable |
| aestronglyMeasurable_const |
| exact Filter.Eventually.of_forall (fun U => by |
| change ‖(Phi U v).toReal‖ = ‖(1 : ℝ)‖ |
| rw [norm_spinToReal, norm_one]) |
| rw [heq, eLpNorm_const' _ (by norm_num) (by norm_num)] |
| simp |
| |
| lemma integral_spinLp {theta : ℝ} (hPhi : Measurable Phi) |
| (hlaw : HasFreeIsingLaw theta (Measure.map Phi (labelMeasure d))) |
| (v : TreeVertex d) : ∫ U, spinLp hPhi v U ∂labelMeasure d = 0 := by |
| exact (integral_congr_ae (coe_spinLp hPhi v)).trans |
| (hlaw.integral_factor_spin hPhi v) |
| |
| lemma spinLp_mem_centered {theta : ℝ} (hPhi : Measurable Phi) |
| (hlaw : HasFreeIsingLaw theta (Measure.map Phi (labelMeasure d))) |
| (v : TreeVertex d) : spinLp hPhi v ∈ centeredL2 (labelMeasure d) := by |
| rw [mem_centeredL2_iff] |
| exact integral_spinLp hPhi hlaw v |
| |
| def centeredSpinLp {theta : ℝ} (hPhi : Measurable Phi) |
| (hlaw : HasFreeIsingLaw theta (Measure.map Phi (labelMeasure d))) |
| (v : TreeVertex d) : centeredL2 (labelMeasure d) := |
| ⟨spinLp hPhi v, spinLp_mem_centered hPhi hlaw v⟩ |
| |
| @[simp] lemma centeredSpinLp_coe {theta : ℝ} (hPhi : Measurable Phi) |
| (hlaw : HasFreeIsingLaw theta (Measure.map Phi (labelMeasure d))) (v : TreeVertex d) : |
| (centeredSpinLp hPhi hlaw v : Lp ℝ 2 (labelMeasure d)) = spinLp hPhi v := rfl |
| |
| @[simp] lemma norm_centeredSpinLp {theta : ℝ} (hPhi : Measurable Phi) |
| (hlaw : HasFreeIsingLaw theta (Measure.map Phi (labelMeasure d))) (v : TreeVertex d) : |
| ‖centeredSpinLp hPhi hlaw v‖ = 1 := norm_spinLp hPhi v |
| |
| lemma koopman_spinLp (hPhi : Measurable Phi) |
| (heq : ∀ g : TreeAut d, |
| ∀ᵐ U ∂labelMeasure d, Phi (relabel g U) = relabel g (Phi U)) |
| (g : TreeAut d) (v : TreeVertex d) : |
| koopmanL2 (relabel g) (measurePreserving_relabel g) (spinLp hPhi v) = |
| spinLp hPhi (g.toEquiv.symm v) := by |
| change Lp.compMeasurePreserving (relabel g) (measurePreserving_relabel g) |
| ((memLp_field hPhi v 2).toLp _) = _ |
| rw [Lp.toLp_compMeasurePreserving] |
| apply MemLp.toLp_congr |
| filter_upwards [heq g] with U hU |
| exact congrArg IsingSpin.toReal (congrFun hU v) |
| |
| lemma inner_spinLp (hPhi : Measurable Phi) (u v : TreeVertex d) : |
| inner ℝ (spinLp hPhi u) (spinLp hPhi v) = |
| ∫ U, field Phi U u * field Phi U v ∂labelMeasure d := by |
| rw [L2.inner_def] |
| apply integral_congr_ae |
| filter_upwards [coe_spinLp hPhi u, coe_spinLp hPhi v] with U hu hv |
| rw [hu, hv] |
| simp [mul_comm] |
| |
| lemma inner_spinLp_root {theta : ℝ} (hPhi : Measurable Phi) |
| (hlaw : HasFreeIsingLaw theta (Measure.map Phi (labelMeasure d))) |
| (hlo : -1 ≤ theta) (hhi : theta ≤ 1) (v : TreeVertex d) : |
| inner ℝ (spinLp hPhi (ConnectedHull.root d)) (spinLp hPhi v) = |
| theta ^ v.1.length := by |
| rw [inner_spinLp] |
| simpa only [field, IsingSpin.toReal_eq_spinSign] using |
| hlaw.factor_root_correlation hPhi hlo hhi v |
| |
| lemma inner_centeredSpinLp_root {theta : ℝ} (hPhi : Measurable Phi) |
| (hlaw : HasFreeIsingLaw theta (Measure.map Phi (labelMeasure d))) |
| (hlo : -1 ≤ theta) (hhi : theta ≤ 1) (v : TreeVertex d) : |
| inner ℝ (centeredSpinLp hPhi hlaw (ConnectedHull.root d)) |
| (centeredSpinLp hPhi hlaw v) = theta ^ v.1.length := |
| inner_spinLp_root hPhi hlaw hlo hhi v |
| |
| lemma inner_koopman_spinLp_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) (g : TreeAut d) : |
| inner ℝ (spinLp hPhi (ConnectedHull.root d)) |
| (koopmanL2 (relabel g) (measurePreserving_relabel g) |
| (spinLp hPhi (ConnectedHull.root d))) = |
| theta ^ (g.toEquiv.symm (ConnectedHull.root d)).1.length := by |
| rw [koopman_spinLp hPhi heq] |
| exact inner_spinLp_root hPhi hlaw hlo hhi _ |
| |
| lemma centeredAverage_centeredSpinLp {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) : |
| TreeKoopman.centeredAverage d (centeredSpinLp hPhi hlaw v) = |
| (d : ℝ)⁻¹ • ∑ c : Fin d, centeredSpinLp hPhi hlaw (prefixFlip c v) := by |
| apply Subtype.ext |
| simp only [TreeKoopman.centeredAverage_apply_coe, Submodule.coe_smul, |
| Submodule.coe_sum, centeredSpinLp_coe] |
| exact TreeKoopman.average_toLp_eq (field Phi) (equivariant_field heq) |
| (fun v => memLp_field hPhi v 2) v |
| |
| end Problem367.FactorObservable |
| |
| namespace Problem367 |
| |
| theorem IsFactorOfIID.exists_centered_observable {d : ℕ} {theta : ℝ} |
| (hfactor : IsFactorOfIID d theta) (hlo : -1 ≤ theta) (hhi : theta ≤ 1) : |
| ∃ f : centeredL2 (labelMeasure d), ‖f‖ = 1 ∧ |
| ∀ g : TreeAut d, |
| inner ℝ (f : Lp ℝ 2 (labelMeasure d)) |
| (koopmanL2 (relabel g) (measurePreserving_relabel g) |
| (f : Lp ℝ 2 (labelMeasure d))) = |
| theta ^ (g.toEquiv.symm (ConnectedHull.root d)).1.length := by |
| obtain ⟨Phi, hPhi, hlaw, heq⟩ := hfactor |
| refine ⟨FactorObservable.centeredSpinLp hPhi hlaw (ConnectedHull.root d), |
| FactorObservable.norm_centeredSpinLp hPhi hlaw _, ?_⟩ |
| intro g |
| exact FactorObservable.inner_koopman_spinLp_root hPhi hlaw heq hlo hhi g |
| |
| end Problem367 |
| |
| end |
| |
| end OAI |
| |
