| import OAI.Probability.FreeIsing.Energy.DiscrepancyRowEnergy | |
| import OAI.Probability.FreeIsing.Gibbs.CovarianceL2Limit | |
| import OAI.Probability.FreeIsing.Stochastic.FiniteIsingBrownianEnergy | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Filter Set | |
| open scoped Topology NNReal ENNReal | |
| namespace Problem367 | |
| def brownianIsingObservationR {d : ℕ} {Ω : Type*} | |
| (B : TreeVertex d → ℝ≥0 → Ω → ℝ) | |
| (p : ℝ × ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω))) : TreeVertex d → ℝ := | |
| fun i => p.1 * spinSign (p.2.1 i) + B i p.1.toNNReal (p.2.2 i) | |
| lemma measurable_brownianIsingObservationR | |
| {d : ℕ} {Ω : Type*} [MeasurableSpace Ω] | |
| {B : TreeVertex d → ℝ≥0 → Ω → ℝ} | |
| (hmB : ∀ i t, Measurable (B i t)) | |
| (hcB : ∀ i omega, Continuous (fun t => B i t omega)) : | |
| Measurable (brownianIsingObservationR B) := by | |
| apply Measurable.of_eval | |
| intro i | |
| apply measurable_uncurry_of_continuous_of_measurable | |
| (u := fun (t : ℝ) (p : (TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) => | |
| t * spinSign (p.1 i) + B i t.toNNReal (p.2 i)) | |
| · intro p | |
| exact (continuous_id.mul continuous_const).add | |
| ((hcB i (p.2 i)).comp continuous_real_toNNReal) | |
| · intro t | |
| exact (measurable_const.mul ((measurable_of_finite spinSign).comp | |
| ((measurable_pi_apply i).comp measurable_fst))).add | |
| ((hmB i t.toNNReal).comp ((measurable_pi_apply i).comp measurable_snd)) | |
| lemma finiteGibbsCovariance_eq_finiteIsingBrownianCovariance | |
| {d : ℕ} {Ω : Type*} (beta : ℝ) (v : TreeVertex d) (n : ℕ) | |
| (B : TreeVertex d → ℝ≥0 → Ω → ℝ) | |
| (p : ℝ × ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω))) | |
| (u : connectedExhaustion v n) : | |
| finiteGibbsCovariance beta v u n (brownianIsingObservationR B p) = | |
| finiteIsingBrownianCovariance beta (connectedExhaustion v n) | |
| ⟨v, mem_connectedExhaustion v n⟩ u B p.1 p.2 := by | |
| classical | |
| simp only [finiteGibbsCovariance, dite_eq_left u.property] | |
| rfl | |
| theorem HasFreeIsingLaw.finiteGibbsCovariance_timeEnergy_le_one | |
| {d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω] | |
| {μ : Measure (TreeVertex d → IsingSpin)} | |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) | |
| (ν : TreeVertex d → Measure Ω) {B : TreeVertex d → ℝ≥0 → Ω → ℝ} | |
| (hB : ∀ i, IsPreBrownianReal (B i) (ν i)) | |
| (hmB : ∀ i t, Measurable (B i t)) | |
| (hcB : ∀ i omega, Continuous (fun t => B i t omega)) | |
| (v : TreeVertex d) (n : ℕ) {T : ℝ} (hT : 0 ≤ T) : | |
| (∫ p, ∑ u ∈ connectedExhaustion v n, | |
| finiteGibbsCovariance beta v u n (brownianIsingObservationR B p) ^ 2 | |
| ∂(volume.restrict (Ioc (0 : ℝ) T)).prod (μ.prod (Measure.infinitePi ν))) ≤ 1 := by | |
| classical | |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure | |
| have (index : TreeVertex d) : IsProbabilityMeasure (ν index) := | |
| (hB index).isGaussianProcess.isProbabilityMeasure | |
| let D := connectedExhaustion v n | |
| let ρ := μ.prod (Measure.infinitePi ν) | |
| let τ : Measure ℝ := volume.restrict (Ioc 0 T) | |
| have hX := measurable_brownianIsingObservationR hmB hcB | |
| have hi (u : TreeVertex d) : Integrable | |
| (fun p => finiteGibbsCovariance beta v u n (brownianIsingObservationR B p) ^ 2) | |
| (τ.prod ρ) := by | |
| apply Integrable.of_bound | |
| (((continuous_finiteGibbsCovariance beta v u n).measurable.comp hX).pow_const 2).aestronglyMeasurable 4 | |
| filter_upwards with p | |
| have h := finiteGibbsCovariance_abs_le_two beta v u n (brownianIsingObservationR B p) | |
| rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] | |
| have hb := (sq_le_sq₀ (abs_nonneg _) (by norm_num : (0 : ℝ) ≤ 2)).mpr h | |
| norm_num only [sq_abs] at hb | |
| simpa only [Function.comp_def] using hb | |
| have hsum : Integrable (fun p => ∑ u ∈ D, | |
| finiteGibbsCovariance beta v u n (brownianIsingObservationR B p) ^ 2) (τ.prod ρ) := | |
| integrable_finsetSum D (fun u _ => hi u) | |
| have heq (p : ℝ × ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω))) : | |
| (∑ u ∈ D, finiteGibbsCovariance beta v u n (brownianIsingObservationR B p) ^ 2) = | |
| ∑ u : D, finiteIsingBrownianCovariance beta D | |
| ⟨v, mem_connectedExhaustion v n⟩ u B p.1 p.2 ^ 2 := by | |
| rw [← Finset.sum_attach D] | |
| apply Finset.sum_congr rfl | |
| intro u _ | |
| rw [finiteGibbsCovariance_eq_finiteIsingBrownianCovariance] | |
| have he := hμ.integral_finiteIsingBrownianEnergy ν hB hmB D | |
| (connectedExhaustion_connected v n) ⟨v, mem_connectedExhaustion v n⟩ hT | |
| dsimp only at he | |
| calc | |
| (∫ p, ∑ u ∈ D, | |
| finiteGibbsCovariance beta v u n (brownianIsingObservationR B p) ^ 2 ∂τ.prod ρ) = | |
| ∫ t, ∫ p, ∑ u ∈ D, | |
| finiteGibbsCovariance beta v u n (brownianIsingObservationR B (t, p)) ^ 2 ∂ρ ∂τ := | |
| integral_prod _ hsum | |
| _ = ∫ t in (0 : ℝ)..T, ∫ p, ∑ u : D, | |
| finiteIsingBrownianCovariance beta D ⟨v, mem_connectedExhaustion v n⟩ u B t p ^ 2 ∂ρ := by | |
| rw [intervalIntegral.integral_of_le hT] | |
| apply integral_congr_ae | |
| filter_upwards with t | |
| apply integral_congr_ae | |
| filter_upwards with p | |
| exact heq (t, p) | |
| _ ≤ 1 := he.2.2.2 | |
| theorem HasFreeIsingLaw.finiteGibbsCovariance_tendsto_timeSquareIntegral | |
| {d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω] | |
| {μ : Measure (TreeVertex d → IsingSpin)} | |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) | |
| (ν : TreeVertex d → Measure Ω) {B : TreeVertex d → ℝ≥0 → Ω → ℝ} | |
| (hB : ∀ i, IsPreBrownianReal (B i) (ν i)) | |
| (hmB : ∀ i t, Measurable (B i t)) | |
| (hcB : ∀ i omega, Continuous (fun t => B i t omega)) | |
| (v w : TreeVertex d) {T : ℝ} : | |
| Tendsto (fun n => ∫ p, | |
| finiteGibbsCovariance beta v w n (brownianIsingObservationR B p) ^ 2 | |
| ∂(volume.restrict (Ioc (0 : ℝ) T)).prod (μ.prod (Measure.infinitePi ν))) atTop | |
| (𝓝 (∫ p, infiniteGibbsCovariance beta v w (brownianIsingObservationR B p) ^ 2 | |
| ∂(volume.restrict (Ioc (0 : ℝ) T)).prod (μ.prod (Measure.infinitePi ν)))) := by | |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure | |
| have (index : TreeVertex d) : IsProbabilityMeasure (ν index) := | |
| (hB index).isGaussianProcess.isProbabilityMeasure | |
| let ρ := μ.prod (Measure.infinitePi ν) | |
| let τ : Measure ℝ := volume.restrict (Ioc 0 T) | |
| have hX := measurable_brownianIsingObservationR hmB hcB | |
| have hm (n : ℕ) := ((continuous_finiteGibbsCovariance beta v w n).measurable.comp hX) | |
| have hy := (measurable_infiniteGibbsCovariance beta v w).comp hX | |
| have hlim : ∀ᵐ p ∂τ.prod ρ, | |
| Tendsto (fun n => finiteGibbsCovariance beta v w n (brownianIsingObservationR B p)) atTop | |
| (𝓝 (infiniteGibbsCovariance beta v w (brownianIsingObservationR B p))) := by | |
| apply (Measure.ae_prod_iff_ae_ae (measurableSet_tendsto_fun hm hy)).mpr | |
| filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht | |
| have h := hμ.finiteGibbsCovariance_brownian_tendsto_ae ν hB hmB t.toNNReal v w | |
| unfold brownianIsingObservationR | |
| simpa only [Function.comp_def, ρ, Real.coe_toNNReal t ht.1.le] using h | |
| apply tendsto_integral_of_dominated_convergence (fun _ => (4 : ℝ)) | |
| (fun n => ((hm n).pow_const 2).aestronglyMeasurable) (integrable_const 4) | |
| · intro n | |
| filter_upwards with p | |
| have h := finiteGibbsCovariance_abs_le_two beta v w n (brownianIsingObservationR B p) | |
| rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] | |
| have hb := (sq_le_sq₀ (abs_nonneg _) (by norm_num : (0 : ℝ) ≤ 2)).mpr h | |
| norm_num only [sq_abs] at hb | |
| simpa only [Function.comp_def] using hb | |
| · exact hlim.mono (fun p hp => hp.pow 2) | |
| theorem HasFreeIsingLaw.infiniteGibbsCovariance_timeEnergy | |
| {d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω] | |
| {μ : Measure (TreeVertex d → IsingSpin)} | |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) | |
| (ν : TreeVertex d → Measure Ω) {B : TreeVertex d → ℝ≥0 → Ω → ℝ} | |
| (hB : ∀ i, IsPreBrownianReal (B i) (ν i)) | |
| (hmB : ∀ i t, Measurable (B i t)) | |
| (hcB : ∀ i omega, Continuous (fun t => B i t omega)) | |
| (v : TreeVertex d) {T : ℝ} (hT : 0 ≤ T) : | |
| let P := (volume.restrict (Ioc (0 : ℝ) T)).prod (μ.prod (Measure.infinitePi ν)) | |
| let C := fun w p => infiniteGibbsCovariance beta v w (brownianIsingObservationR B p) | |
| Summable (fun w => ∫ p, C w p ^ 2 ∂P) ∧ | |
| (∑' w, ∫ p, C w p ^ 2 ∂P) ≤ 1 ∧ | |
| (∀ᵐ p ∂P, Summable (fun w => C w p ^ 2)) ∧ | |
| Integrable (fun p => ∑' w, C w p ^ 2) P ∧ | |
| (∫ p, ∑' w, C w p ^ 2 ∂P) ≤ 1 := by | |
| classical | |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure | |
| have (index : TreeVertex d) : IsProbabilityMeasure (ν index) := | |
| (hB index).isGaussianProcess.isProbabilityMeasure | |
| let P := (volume.restrict (Ioc (0 : ℝ) T)).prod (μ.prod (Measure.infinitePi ν)) | |
| let C := fun w p => infiniteGibbsCovariance beta v w (brownianIsingObservationR B p) | |
| let Cn := fun n w p => finiteGibbsCovariance beta v w n (brownianIsingObservationR B p) | |
| have hX := measurable_brownianIsingObservationR hmB hcB | |
| have hm (w : TreeVertex d) : Measurable (C w) := | |
| (measurable_infiniteGibbsCovariance beta v w).comp hX | |
| have hmn (n : ℕ) (w : TreeVertex d) : Measurable (Cn n w) := | |
| (continuous_finiteGibbsCovariance beta v w n).measurable.comp hX | |
| have hi (w : TreeVertex d) : Integrable (fun p => C w p ^ 2) P := by | |
| apply Integrable.of_bound ((hm w).pow_const 2).aestronglyMeasurable 4 | |
| filter_upwards with p | |
| have h := infiniteGibbsCovariance_abs_le_two beta v w (brownianIsingObservationR B p) | |
| rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] | |
| have hb := (sq_le_sq₀ (abs_nonneg _) (by norm_num : (0 : ℝ) ≤ 2)).mpr h | |
| norm_num only [sq_abs] at hb | |
| exact hb | |
| have hin (n : ℕ) (w : TreeVertex d) : Integrable (fun p => Cn n w p ^ 2) P := by | |
| apply Integrable.of_bound ((hmn n w).pow_const 2).aestronglyMeasurable 4 | |
| filter_upwards with p | |
| have h := finiteGibbsCovariance_abs_le_two beta v w n (brownianIsingObservationR B p) | |
| rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] | |
| have hb := (sq_le_sq₀ (abs_nonneg _) (by norm_num : (0 : ℝ) ≤ 2)).mpr h | |
| norm_num only [sq_abs] at hb | |
| exact hb | |
| have hevent (S : Finset (TreeVertex d)) : | |
| ∀ᶠ n in atTop, S ⊆ connectedExhaustion v n := by | |
| induction S using Finset.induction_on with | |
| | empty => exact Eventually.of_forall fun _ => Finset.empty_subset _ | |
| | @insert w S hw ih => | |
| filter_upwards [eventually_mem_connectedExhaustion v w, ih] with n hn hS | |
| exact Finset.insert_subset hn hS | |
| have hpartial (S : Finset (TreeVertex d)) (n : ℕ) | |
| (hS : S ⊆ connectedExhaustion v n) : | |
| (∑ w ∈ S, ∫ p, Cn n w p ^ 2 ∂P) ≤ 1 := by | |
| rw [← integral_finsetSum S (fun w _ => hin n w)] | |
| apply le_trans (integral_mono_ae | |
| (integrable_finsetSum S (fun w _ => hin n w)) | |
| (integrable_finsetSum (connectedExhaustion v n) (fun w _ => hin n w)) | |
| (Eventually.of_forall (fun p => | |
| Finset.sum_le_sum_of_subset_of_nonneg hS (fun w _ _ => sq_nonneg (Cn n w p))))) | |
| exact hμ.finiteGibbsCovariance_timeEnergy_le_one ν hB hmB hcB v n hT | |
| have hbound (S : Finset (TreeVertex d)) : | |
| (∑ w ∈ S, ∫ p, C w p ^ 2 ∂P) ≤ 1 := by | |
| apply le_of_tendsto | |
| (tendsto_finsetSum S (fun w _ => | |
| hμ.finiteGibbsCovariance_tendsto_timeSquareIntegral ν hB hmB hcB v w)) | |
| exact (hevent S).mono (fun n hn => hpartial S n hn) | |
| have hs : Summable (fun w => ∫ p, C w p ^ 2 ∂P) := | |
| summable_of_sum_le (fun w => integral_nonneg (fun p => sq_nonneg (C w p))) hbound | |
| have hle : (∑' w, ∫ p, C w p ^ 2 ∂P) ≤ 1 := hs.tsum_le_of_sum_le hbound | |
| have hrow := DiscrepancyEnergy.sq_row_integrable_of_summable_integrals hi hs | |
| have hident := DiscrepancyEnergy.hasSum_integral_sq_row | |
| (fun w => (hm w).aestronglyMeasurable) hrow.1 hrow.2 | |
| dsimp only | |
| refine ⟨hs, hle, hrow.1, hrow.2, ?_⟩ | |
| rw [← hident.tsum_eq] | |
| exact hle | |
| end Problem367 | |
| end | |
| end OAI | |