| import OAI.Probability.FreeIsing.Energy.NoisyLocalCovariance |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory |
| namespace Problem367.NoisyLawSpectral |
| open PlantedStationarity NoisyLocalCovariance |
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| variable {d : ℕ} {A B : Type*} [MeasurableSpace A] [MeasurableSpace B] |
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| abbrev law (μ : Measure (TreeVertex d → IsingSpin)) (ν : Measure A) |
| (O : IsingSpin → A → B) : Measure (TreeVertex d → B) := |
| (μ.prod (noiseLaw ν)).map (coordinateMap O) |
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| theorem local_koopman_far_decay {theta : ℝ} |
| {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ] |
| (hμ : HasFreeIsingLaw theta μ) (h0 : 0 ≤ theta) (h1 : theta ≤ 1) |
| (ν : Measure A) [IsProbabilityMeasure ν] |
| {O : IsingSpin → A → B} (hO : Measurable (Function.uncurry O)) |
| (s : Finset (TreeVertex d)) (f : Lp ℝ 2 (law μ ν O)) |
| (F : (s → B) → ℝ) (hF : Measurable F) |
| (hf : f =ᵐ[law μ ν O] fun x => F (fun i => x i)) |
| (hzero : ∫ x, f x ∂law μ ν O = 0) : |
| ∃ N : ℕ, ∀ g : TreeAut d, N < (g.toEquiv TreeEscape.root).val.length → |
| |inner ℝ f (koopmanL2 (relabel g) |
| (measurePreserving_coordinateMap_relabel hμ ν O hO g) f)| ≤ |
| ‖f‖^2 * theta^((g.toEquiv TreeEscape.root).val.length - N) := by |
| let f0 : (TreeVertex d → B) → ℝ := fun x => F (fun i => x i) |
| have hm0 : Measurable f0 := hF.comp |
| (Measurable.of_eval (fun i => measurable_pi_apply (i : TreeVertex d))) |
| have hf0 : MemLp f0 2 (law μ ν O) := (memLp_congr_ae hf).mp (Lp.memLp f) |
| have hmO := measurable_coordinateMap (d := d) hO |
| have hmp : MeasurePreserving (coordinateMap (d := d) O) |
| (μ.prod (noiseLaw ν)) (law μ ν O) := ⟨hmO, rfl⟩ |
| have h2 := hf0.comp_measurePreserving hmp |
| have hz0 : (∫ x, f0 x ∂law μ ν O) = 0 := (integral_congr_ae hf).symm.trans hzero |
| have hz : (∫ p, f0 (coordinateMap O p) ∂μ.prod (noiseLaw ν)) = 0 := |
| (integral_map hmO.aemeasurable hm0.aestronglyMeasurable).symm.trans hz0 |
| have hn0 : (∫ x, (f0 x)^2 ∂law μ ν O) = ‖f‖^2 := by |
| rw [← real_inner_self_eq_norm_sq f, L2.inner_def] |
| apply integral_congr_ae |
| filter_upwards [hf] with x hx |
| simp [hx, f0, pow_two] |
| have hn : (∫ p, (f0 (coordinateMap O p))^2 ∂μ.prod (noiseLaw ν)) = ‖f‖^2 := |
| (integral_map hmO.aemeasurable (hm0.pow_const 2).aestronglyMeasurable).symm.trans hn0 |
| let R := s.sup (fun v => v.val.length) |
| refine ⟨R + R, fun g hg => ?_⟩ |
| have hb := NoisyLocalCovariance.abs_correlation_far hμ h0 h1 ν hO s hF h2 hz g hg |
| change |∫ p, f0 (coordinateMap O p) * |
| f0 (relabel g (coordinateMap O p)) ∂μ.prod (noiseLaw ν)| ≤ |
| (∫ p, (f0 (coordinateMap O p))^2 ∂μ.prod (noiseLaw ν)) * |
| theta ^ ((g.toEquiv TreeEscape.root).val.length - (R + R)) at hb |
| rw [hn] at hb |
| rw [TreeLawKoopman.inner_koopman_eq_integral_of_local (law μ ν O) |
| (measurePreserving_coordinateMap_relabel hμ ν O hO) s f F hf g] |
| change |∫ x, f0 x * f0 (relabel g x) ∂law μ ν O| ≤ _ |
| have heq : (∫ x, f0 x * f0 (relabel g x) ∂law μ ν O) = |
| ∫ p, f0 (coordinateMap O p) * f0 (relabel g (coordinateMap O p)) |
| ∂μ.prod (noiseLaw ν) := |
| integral_map hmO.aemeasurable |
| (hm0.mul (hm0.comp (measurable_relabel g))).aestronglyMeasurable |
| rw [heq] |
| exact hb |
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| theorem localCriticalDecay {theta : ℝ} |
| {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ] |
| (hμ : HasFreeIsingLaw theta μ) (hd : 3 ≤ d) (h0 : 0 ≤ theta) (h1 : theta ≤ 1) |
| (hthreshold : theta ≤ (Real.sqrt ((d : ℝ) - 1))⁻¹) |
| (ν : Measure A) [IsProbabilityMeasure ν] |
| {O : IsingSpin → A → B} (hO : Measurable (Function.uncurry O)) : |
| TreeLawKoopman.LocalCriticalDecay (law μ ν O) |
| (measurePreserving_coordinateMap_relabel hμ ν O hO) := by |
| apply TreeLawKoopman.localCriticalDecay_of_far_decay (law μ ν O) |
| (measurePreserving_coordinateMap_relabel hμ ν O hO) hd h0 hthreshold |
| intro s f F hF hf hzero |
| exact local_koopman_far_decay hμ h0 h1 ν hO s f F hF hf hzero |
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| theorem centeredAverage_norm_le_critical {theta : ℝ} |
| {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ] |
| (hμ : HasFreeIsingLaw theta μ) (hd : 3 ≤ d) (h0 : 0 ≤ theta) (h1 : theta ≤ 1) |
| (hthreshold : theta ≤ (Real.sqrt ((d : ℝ) - 1))⁻¹) |
| (ν : Measure A) [IsProbabilityMeasure ν] |
| {O : IsingSpin → A → B} (hO : Measurable (Function.uncurry O)) : |
| ‖TreeLawKoopman.centeredAverage (law μ ν O) |
| (measurePreserving_coordinateMap_relabel hμ ν O hO)‖ ≤ |
| 2 * Real.sqrt ((d : ℝ) - 1) / d := by |
| exact TreeLawKoopman.centeredAverage_norm_le (law μ ν O) |
| (measurePreserving_coordinateMap_relabel hμ ν O hO) hd |
| (localCriticalDecay hμ hd h0 h1 hthreshold ν hO) |
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| end Problem367.NoisyLawSpectral |
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| end |
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| end OAI |
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