| import OAI.Probability.FreeIsing.Ising.ContinuousRootResponse |
| import OAI.Probability.FreeIsing.Ising.CopyBrownian |
| import OAI.Probability.FreeIsing.Ising.CopyEnergyIntegrability |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped NNReal Topology |
|
|
| namespace Problem367.IsingContinuousCopyCoefficients |
| open IsingObservationInnovation IsingObservationFactor IsingCopyEnergy |
|
|
| variable {d : ℕ} {beta : ℝ} |
|
|
| def response (hb : 0 ≤ beta) (j : Bool) (v : TreeVertex d) |
| (t : ℝ≥0) (q : Joint d) : ℝ := |
| continuousRootResponse hb v t (projection j q) |
|
|
| def covariance (hb : 0 ≤ beta) (j : Bool) (v : TreeVertex d) |
| (t : ℝ≥0) (q : Joint d) : ℝ := |
| continuousRootCovariance hb v t (projection j q) |
|
|
| lemma projection_measurable (j : Bool) : Measurable (projection (d := d) j) := by |
| cases j |
| · exact measurable_snd.comp measurable_snd |
| · exact measurable_fst.comp measurable_snd |
|
|
| lemma projection_preserving {μ : Measure (TreeVertex d → IsingSpin)} |
| [IsProbabilityMeasure μ] (j : Bool) : |
| MeasurePreserving (projection (d := d) j) (copies beta μ) (observationLaw μ) := by |
| obtain ⟨hl, hr⟩ := IsingCopyStationarity.measurePreserving_observations |
| (beta := beta) (μ := μ) |
| cases j |
| · exact hr |
| · exact hl |
|
|
| lemma field_progressive (j : Bool) (v : TreeVertex d) : |
| IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration |
| (fun t (q : Joint d) => continuousField beta v t (projection j q)) := by |
| have hobs (i : TreeVertex d) : IsStronglyProgressive |
| ContinuousCopyBrownian.joinedFiltration |
| (fun t (q : Joint d) => projection j q i t) := by |
| cases j |
| · exact IsingConditionalCopyBounds.progressive_rightObservation i |
| · exact IsingConditionalCopyBounds.progressive_leftObservation i |
| have hraw := isStronglyProgressive_artanh _ |
| (isStronglyProgressive_infiniteGibbsSpinMean _ beta v _ hobs) |
| exact ProgressiveVersionPushforward.progressive_version hraw |
|
|
| lemma response_progressive (hb : 0 ≤ beta) (j : Bool) (v : TreeVertex d) : |
| IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration (response hb j v) := by |
| intro T |
| let : MeasurableSpace (Joint d) := ContinuousCopyBrownian.joinedFiltration T |
| exact ((continuous_siteRootResponse hb v).measurable.comp |
| (Measurable.of_eval (fun vertex => (field_progressive j vertex T).measurable))).stronglyMeasurable |
|
|
| private lemma continuous_tanh : Continuous Real.tanh := |
| continuous_iff_continuousAt.mpr (fun x => (hasDerivAt_tanh_response x).continuousAt) |
|
|
| lemma covariance_progressive (hb : 0 ≤ beta) (j : Bool) (v : TreeVertex d) : |
| IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration (covariance hb j v) := by |
| intro T |
| exact (stronglyMeasurable_const.sub |
| ((continuous_tanh.comp_stronglyMeasurable |
| (field_progressive j (ConnectedHull.root d) T)).pow 2)).mul |
| (response_progressive hb j v T) |
|
|
| lemma response_abs_le_one (hb : 0 ≤ beta) (j : Bool) (v : TreeVertex d) |
| (t : ℝ≥0) (q : Joint d) : |response hb j v t q| ≤ 1 := |
| continuousRootResponse_abs_le_one hb v t (projection j q) |
|
|
| lemma covariance_abs_le_one (hb : 0 ≤ beta) (j : Bool) (v : TreeVertex d) |
| (t : ℝ≥0) (q : Joint d) : |covariance hb j v t q| ≤ 1 := |
| continuousRootCovariance_abs_le_one hb v t (projection j q) |
|
|
| lemma covariance_sq_le_response_sq (hb : 0 ≤ beta) (j : Bool) (v : TreeVertex d) |
| (t : ℝ≥0) (q : Joint d) : covariance hb j v t q ^ 2 ≤ response hb j v t q ^ 2 := by |
| have hz := Real.tanh_sq_lt_one (continuousField beta (ConnectedHull.root d) t (projection j q)) |
| have hn := sq_nonneg (Real.tanh (continuousField beta (ConnectedHull.root d) t (projection j q))) |
| change ((1 - Real.tanh (continuousField beta (ConnectedHull.root d) t (projection j q)) ^ 2) * |
| response hb j v t q) ^ 2 ≤ _ |
| rw [mul_pow] |
| exact mul_le_of_le_one_left (sq_nonneg _) (by nlinarith) |
|
|
| variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ] |
|
|
| lemma response_continuous_ae (hd : 3 ≤ d) |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) |
| (j : Bool) (v : TreeVertex d) : |
| ∀ᵐ q ∂copies beta μ, Continuous (fun t => response hb j v t q) := |
| (projection_preserving (beta := beta) (μ := μ) j).quasiMeasurePreserving.ae |
| (continuousRootResponse_continuous_ae hd hμ hb hcrit v) |
|
|
| lemma covariance_continuous_ae (hd : 3 ≤ d) |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) |
| (j : Bool) (v : TreeVertex d) : |
| ∀ᵐ q ∂copies beta μ, Continuous (fun t => covariance hb j v t q) := |
| (projection_preserving (beta := beta) (μ := μ) j).quasiMeasurePreserving.ae |
| (continuousRootCovariance_continuous_ae hd hμ hb hcrit v) |
|
|
| lemma response_eq_raw_ae (hd : 3 ≤ d) |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) |
| (j : Bool) (v : TreeVertex d) (t : ℝ≥0) : |
| response hb j v t =ᵐ[copies beta μ] fun q => |
| infiniteGibbsResponse beta (ConnectedHull.root d) v (fun i => projection j q i t) := |
| (projection_preserving (beta := beta) (μ := μ) j).quasiMeasurePreserving.ae |
| (continuousRootResponse_eq_response_ae hd hμ hb hcrit v t) |
|
|
| lemma covariance_eq_raw_ae (hd : 3 ≤ d) |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) |
| (j : Bool) (v : TreeVertex d) (t : ℝ≥0) : |
| covariance hb j v t =ᵐ[copies beta μ] fun q => |
| infiniteGibbsCovariance beta (ConnectedHull.root d) v (fun i => projection j q i t) := |
| (projection_preserving (beta := beta) (μ := μ) j).quasiMeasurePreserving.ae |
| (continuousRootCovariance_eq_covariance_ae hd hμ hb hcrit v t) |
|
|
| private lemma real_measurable {X : ℝ≥0 → Joint d → ℝ} |
| (hX : IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration X) : |
| Measurable (fun p : ℝ × Joint d => X p.1.toNNReal p.2) := |
| (stronglyMeasurable_joint_of_isStronglyProgressive hX).measurable.comp |
| ((measurable_real_toNNReal.comp measurable_fst).prodMk measurable_snd) |
|
|
| private lemma sq_time_integrable {X : ℝ≥0 → Joint d → ℝ} |
| (hX : IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration X) |
| (hb : ∀ t q, |X t q| ≤ 1) (T : ℝ≥0) : |
| Integrable (fun p : ℝ × Joint d => X p.1.toNNReal p.2 ^ 2) |
| ((volume.restrict (Ioc (0 : ℝ) (T : ℝ))).prod (copies beta μ)) := by |
| have hp : IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration |
| (fun t q => X t q ^ 2) := fun t => (hX t).pow 2 |
| have h := integrable_nnreal_process_interval_of_progressive_bounded |
| (P := copies beta μ) hp 1 |
| (fun t => Eventually.of_forall fun q => by |
| rw [abs_of_nonneg (sq_nonneg _)] |
| nlinarith [hb t q, neg_abs_le (X t q), le_abs_self (X t q)]) 0 T |
| simpa only [NNReal.coe_zero] using h |
|
|
| theorem response_row_energy (hd : 3 ≤ d) |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) |
| (j : Bool) (T : ℝ≥0) : |
| (∀ᵐ p ∂((volume.restrict (Ioc (0 : ℝ) (T : ℝ))).prod (copies beta μ)), |
| Summable (fun v => response hb j v p.1.toNNReal p.2 ^ 2)) ∧ |
| Integrable (fun p : ℝ × Joint d => ∑' v, response hb j v p.1.toNNReal p.2 ^ 2) |
| ((volume.restrict (Ioc (0 : ℝ) (T : ℝ))).prod (copies beta μ)) := by |
| have he (v : TreeVertex d) : |
| (fun p : ℝ × Joint d => response hb j v p.1.toNNReal p.2) =ᵐ[ |
| (volume.restrict (Ioc (0 : ℝ) (T : ℝ))).prod (copies beta μ)] |
| fun p => responseRow beta (ConnectedHull.root d) v (timeProjection j p) := by |
| apply (Measure.ae_prod_iff_ae_ae (measurableSet_eq_fun |
| (real_measurable (response_progressive hb j v)) |
| ((measurable_responseRow beta (ConnectedHull.root d) v).comp |
| (measurable_fst.prodMk ((projection_measurable j).comp measurable_snd))))).2 |
| exact Eventually.of_forall fun t => response_eq_raw_ae hd hμ hb hcrit j v t.toNNReal |
| have hall := ae_all_iff.mpr he |
| have hraw := copy_row_energy_integrable hμ hb (ConnectedHull.root d) j T.coe_nonneg |
| constructor |
| · filter_upwards [hall, hraw.1] with p hp hs |
| simpa only [hp] using hs |
| · apply hraw.2.1.congr |
| filter_upwards [hall] with p hp |
| exact tsum_congr (fun v => congrArg (fun x : ℝ => x ^ 2) (hp v).symm) |
| |
| theorem response_energy_summable (hd : 3 ≤ d) |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) |
| (j : Bool) (T : ℝ≥0) : |
| Summable (fun v => ∫ s in (0 : ℝ)..(T : ℝ), ∫ q, |
| response hb j v s.toNNReal q ^ 2 ∂copies beta μ) := by |
| obtain ⟨hs, hi⟩ := response_row_energy hd hμ hb hcrit j T |
| have h := (DiscrepancyEnergy.hasSum_integral_sq_row |
| (fun v => (real_measurable (response_progressive hb j v)).aestronglyMeasurable) hs hi).summable |
| have he (v : TreeVertex d) : |
| (∫ p : ℝ × Joint d, response hb j v p.1.toNNReal p.2 ^ 2 |
| ∂(volume.restrict (Ioc (0 : ℝ) (T : ℝ))).prod (copies beta μ)) = |
| ∫ s in (0 : ℝ)..(T : ℝ), ∫ q, response hb j v s.toNNReal q ^ 2 ∂copies beta μ := by |
| rw [integral_prod _ (sq_time_integrable (response_progressive hb j v) |
| (response_abs_le_one hb j v) T), intervalIntegral.integral_of_le T.coe_nonneg] |
| simpa only [he] using h |
| |
| theorem covariance_energy_summable (hd : 3 ≤ d) |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) |
| (j : Bool) (T : ℝ≥0) : |
| Summable (fun v => ∫ s in (0 : ℝ)..(T : ℝ), ∫ q, |
| covariance hb j v s.toNNReal q ^ 2 ∂copies beta μ) := by |
| apply Summable.of_nonneg_of_le |
| (fun v => intervalIntegral.integral_nonneg T.coe_nonneg |
| (fun s _ => integral_nonneg (fun q => sq_nonneg _))) |
| _ (response_energy_summable hd hμ hb hcrit j T) |
| intro v |
| have hC := sq_time_integrable (covariance_progressive hb j v) |
| (covariance_abs_le_one hb j v) T (beta := beta) (μ := μ) |
| have hR := sq_time_integrable (response_progressive hb j v) |
| (response_abs_le_one hb j v) T (beta := beta) (μ := μ) |
| rw [intervalIntegral.integral_of_le T.coe_nonneg, intervalIntegral.integral_of_le T.coe_nonneg, |
| ← integral_prod _ hC, ← integral_prod _ hR] |
| exact integral_mono hC hR (fun p => covariance_sq_le_response_sq hb j v p.1.toNNReal p.2) |
| |
| end Problem367.IsingContinuousCopyCoefficients |
| |
| end |
| |
| end OAI |
| |
