| import OAI.Probability.FreeIsing.Energy.FreeLawLocalCovariance |
| import OAI.Probability.FreeIsing.Trees.LawDecay |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory |
| namespace Problem367 |
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| theorem HasFreeIsingLaw.local_koopman_far_decay {d : ℕ} {theta : ℝ} |
| {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ] |
| (hμ : HasFreeIsingLaw theta μ) (h0 : 0 ≤ theta) (h1 : theta ≤ 1) |
| (s : Finset (TreeVertex d)) (f : Lp ℝ 2 μ) (F : (s → IsingSpin) → ℝ) |
| (hF : Measurable F) (hf : f =ᵐ[μ] fun x => F (fun i => x i)) |
| (hzero : ∫ x, f x ∂μ = 0) : |
| ∃ N : ℕ, ∀ g : TreeAut d, N < (g.toEquiv TreeEscape.root).val.length → |
| |inner ℝ f (koopmanL2 (relabel g) (hμ.measurePreserving_relabel g) f)| ≤ |
| ‖f‖^2 * theta^((g.toEquiv TreeEscape.root).val.length - N) := by |
| let R := s.sup (fun v => v.val.length) |
| have hR : ∀ i : s, i.val.val.length ≤ R := fun i => |
| Finset.le_sup (f := fun v : TreeVertex d => v.val.length) i.property |
| let f0 : (TreeVertex d → IsingSpin) → ℝ := fun x => F (fun i => x i) |
| have hf0 : MemLp f0 2 μ := (memLp_congr_ae hf).mp (Lp.memLp f) |
| have hz : ∫ x, f0 x ∂μ = 0 := (integral_congr_ae hf).symm.trans hzero |
| have hn : (∫ x, (f0 x)^2 ∂μ) = ‖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] |
| refine ⟨R + R, fun g hg => ?_⟩ |
| have hdepth : (g.symm.toEquiv TreeEscape.root).val.length = |
| (g.toEquiv TreeEscape.root).val.length := TreeEscape.symm_root_depth g |
| have hgz : ∫ x, f0 (relabel g x) ∂μ = 0 := |
| (hμ.integral_relabel g f0).trans hz |
| have hgn : (∫ x, (f0 (relabel g x))^2 ∂μ) = ‖f‖^2 := |
| (hμ.integral_relabel g (fun x => (f0 x)^2)).trans hn |
| have hgf : MemLp (fun x => f0 (relabel g x)) 2 μ := |
| hf0.comp_measurePreserving (hμ.measurePreserving_relabel g) |
| have hb := hμ.abs_correlation_separated_local_blocks h0 h1 |
| (fun i : s => i.val) (fun i : s => i.val) R hR hR g.symm |
| (hdepth ▸ hg) F F hF hF hz hgz hf0 hgf |
| rw [TreeLawKoopman.inner_koopman_eq_integral_of_local μ |
| hμ.measurePreserving_relabel s f F hf g] |
| change |∫ x, f0 x * f0 (relabel g x) ∂μ| ≤ _ |
| change |∫ x, f0 x * f0 (relabel g x) ∂μ| ≤ |
| Real.sqrt (∫ x, (f0 x)^2 ∂μ) * Real.sqrt (∫ x, (f0 (relabel g x))^2 ∂μ) * |
| theta ^ ((g.symm.toEquiv TreeEscape.root).val.length - (R + R)) at hb |
| simpa only [hn, hgn, Real.sqrt_sq_eq_abs, abs_norm, ← pow_two, hdepth] using hb |
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| theorem HasFreeIsingLaw.localCriticalDecay {d : ℕ} {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) : |
| TreeLawKoopman.LocalCriticalDecay μ hμ.measurePreserving_relabel := by |
| apply TreeLawKoopman.localCriticalDecay_of_far_decay μ hμ.measurePreserving_relabel |
| hd h0 hthreshold |
| intro s f F hF hf hzero |
| exact hμ.local_koopman_far_decay h0 h1 s f F hF hf hzero |
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| theorem HasFreeIsingLaw.centeredAverage_norm_le_critical {d : ℕ} {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) : |
| ‖TreeLawKoopman.centeredAverage μ hμ.measurePreserving_relabel‖ ≤ |
| 2 * Real.sqrt ((d : ℝ) - 1) / d := |
| TreeLawKoopman.centeredAverage_norm_le μ hμ.measurePreserving_relabel hd |
| (hμ.localCriticalDecay hd h0 h1 hthreshold) |
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|
| end Problem367 |
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| end |
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| end OAI |
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