| import OAI.Probability.FreeIsing.Brownian.DriftMoments |
| import OAI.Probability.FreeIsing.Energy.FiniteIsingFieldEnergy |
| import OAI.Probability.FreeIsing.Stochastic.FiniteIsingBrownianEnergy |
| import OAI.Probability.FreeIsing.Trees.GibbsField |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped NNReal ENNReal |
|
|
| namespace Problem367 |
|
|
| local instance finiteIsingBrownianEntropyNonemptySpin : Nonempty IsingSpin := ⟨.neg⟩ |
|
|
| def finiteIsingBrownianEntropy {d : ℕ} {Ω : Type*} |
| (beta : ℝ) (D : Finset (TreeVertex d)) (v : D) |
| (B : TreeVertex d → ℝ≥0 → Ω → ℝ) (t : ℝ) |
| (p : (TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) : ℝ := |
| FinitePosterior.fieldEntropy (finiteIsingEnergy beta D) |
| (fun eta u => spinSign (eta u)) (fun eta => spinSign (eta v)) |
| (finiteIsingBrownianObservation B D t p) |
|
|
| def finiteIsingBrownianNormalizedCovariance {d : ℕ} {Ω : Type*} |
| (beta : ℝ) (D : Finset (TreeVertex d)) (v u : D) |
| (B : TreeVertex d → ℝ≥0 → Ω → ℝ) (t : ℝ) |
| (p : (TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) : ℝ := |
| FinitePosterior.normalizedFieldCovariance (finiteIsingEnergy beta D) |
| (fun eta w => spinSign (eta w)) (fun eta => spinSign (eta v)) |
| (finiteIsingBrownianObservation B D t p) u |
|
|
| theorem finiteIsingBrownianEntropy_nonneg {d : ℕ} {Ω : Type*} |
| (beta : ℝ) (D : Finset (TreeVertex d)) (v : D) |
| (B : TreeVertex d → ℝ≥0 → Ω → ℝ) (t : ℝ) |
| (p : (TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) : |
| 0 ≤ finiteIsingBrownianEntropy beta D v B t p := by |
| exact FinitePosterior.fieldEntropy_nonneg _ _ _ _ |
| (fun eta => by cases eta v <;> simp [spinSign]) |
| ⟨fun _ => .neg, rfl⟩ ⟨fun _ => .pos, rfl⟩ |
|
|
| theorem finiteIsingBrownianEntropy_le_noise {d : ℕ} {Ω : Type*} |
| {beta : ℝ} (hbeta : 0 ≤ beta) (D : Finset (TreeVertex d)) (v : D) |
| (B : TreeVertex d → ℝ≥0 → Ω → ℝ) (t : ℝ≥0) |
| (p : (TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) : |
| finiteIsingBrownianEntropy beta D v B t p ≤ |
| 2 * ((d : ℝ) * beta + (t : ℝ) + |B v t (p.2 v)|) := by |
| let y := finiteIsingBrownianObservation B D t p |
| have he := FinitePosterior.fieldEntropy_le_abs_field (finiteIsingEnergy beta D) |
| (fun (eta : D → IsingSpin) w => spinSign (eta w)) (fun eta => spinSign (eta v)) y |
| (fun eta => by cases eta v <;> simp [spinSign]) |
| ⟨fun _ => .neg, rfl⟩ ⟨fun _ => .pos, rfl⟩ |
| have hb := finite_tree_gibbs_field_uniform_bound D hbeta y v |
| have ho : |y v| ≤ (t : ℝ) + |B v t (p.2 v)| := by |
| change |(t : ℝ) * spinSign (p.1 v) + B v (Real.toNNReal (t : ℝ)) (p.2 v)| ≤ _ |
| rw [Real.toNNReal_coe] |
| calc |
| _ ≤ |(t : ℝ) * spinSign (p.1 v)| + |B v t (p.2 v)| := abs_add_le _ _ |
| _ = _ := by cases p.1 v <;> simp [spinSign, abs_of_nonneg t.coe_nonneg] |
| have hfield : |FinitePosterior.fieldField (finiteIsingEnergy beta D) |
| (fun (eta : D → IsingSpin) w => spinSign (eta w)) (fun eta => spinSign (eta v)) y| ≤ |
| (d : ℝ) * beta + (t : ℝ) + |B v t (p.2 v)| := by |
| have htri := abs_sub_le (FinitePosterior.fieldField (finiteIsingEnergy beta D) |
| (fun (eta : D → IsingSpin) w => spinSign (eta w)) (fun eta => spinSign (eta v)) y) |
| (y v) 0 |
| simp only [sub_zero] at htri |
| change |FinitePosterior.fieldField (finiteIsingEnergy beta D) |
| (fun (eta : D → IsingSpin) w => spinSign (eta w)) |
| (fun eta => spinSign (eta v)) y - y v| ≤ (d : ℝ) * beta at hb |
| linarith |
| exact he.trans (mul_le_mul_of_nonneg_left hfield (by norm_num)) |
|
|
| private theorem integral_abs_gaussian_le_sqrt |
| {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] |
| {X : Ω → ℝ} {t : ℝ≥0} (hX : HasLaw X (gaussianReal 0 t) P) : |
| (∫ ω, |X ω| ∂P) ≤ Real.sqrt (t : ℝ) := by |
| have hX2 : Integrable (fun ω => X ω ^ 2) P := by |
| simpa only [sq_abs] using integrable_abs_pow_of_gaussianLaw hX 2 |
| have htrans := hX.integral_comp (f := fun x : ℝ => x ^ 2) (by fun_prop) |
| have hvar := variance_fun_id_gaussianReal (μ := 0) (v := t) |
| rw [variance_of_integral_eq_zero (μ := gaussianReal 0 t) (X := fun x : ℝ => x) |
| (by fun_prop) (by simp)] at hvar |
| have hsq : (∫ ω, X ω ^ 2 ∂P) = (t : ℝ) := htrans.trans hvar |
| exact (integral_abs_le_sqrt_integral_sq hX.aemeasurable hX2).trans_eq |
| (congrArg Real.sqrt hsq) |
|
|
| theorem finiteIsingBrownianEntropy_moment |
| {d : ℕ} {Ω : Type*} [MeasurableSpace Ω] {beta : ℝ} (hbeta : 0 ≤ beta) |
| (μ : Measure (TreeVertex d → IsingSpin)) [IsProbabilityMeasure 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)) |
| (D : Finset (TreeVertex d)) (v : D) (t : ℝ≥0) : |
| Integrable (finiteIsingBrownianEntropy beta D v B t) (μ.prod (Measure.infinitePi ν)) ∧ |
| (∫ p, finiteIsingBrownianEntropy beta D v B t p ∂μ.prod (Measure.infinitePi ν)) ≤ |
| 2 * ((d : ℝ) * beta + (t : ℝ) + Real.sqrt (t : ℝ)) := by |
| have (i : TreeVertex d) : IsProbabilityMeasure (ν i) := |
| (hB i).isGaussianProcess.isProbabilityMeasure |
| let ρ := μ.prod (Measure.infinitePi ν) |
| let X : ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) → ℝ := |
| fun p => B v t (p.2 v) |
| have hcoord := (measurePreserving_eval_infinitePi ν (v : TreeVertex d)).comp |
| (measurePreserving_snd (μ := μ) (ν := Measure.infinitePi ν)) |
| have hX : HasLaw X (gaussianReal 0 t) ρ := |
| (hB v).hasLaw_eval t |>.comp ⟨hcoord.measurable.aemeasurable, hcoord.map_eq⟩ |
| have hXi : Integrable (fun p => |X p|) ρ := by |
| simpa only [pow_one] using integrable_abs_pow_of_gaussianLaw hX 1 |
| have hY : Measurable (finiteIsingBrownianObservation B D (t : ℝ)) := by |
| apply Measurable.of_eval |
| intro w |
| exact (((measurable_of_finite spinSign).comp |
| ((measurable_pi_apply (w : TreeVertex d)).comp measurable_fst)).const_mul (t : ℝ)).add |
| ((hmB w (t : ℝ).toNNReal).comp ((measurable_pi_apply (w : TreeVertex d)).comp measurable_snd)) |
| have hF : Measurable (finiteIsingBrownianEntropy beta D v B (t : ℝ)) := |
| (FinitePosterior.continuous_fieldEntropy (finiteIsingEnergy beta D) |
| (fun (eta : D → IsingSpin) w => spinSign (eta w)) (fun eta => spinSign (eta v)) |
| (fun eta => by cases eta v <;> simp [spinSign]) |
| ⟨fun _ => .neg, rfl⟩ ⟨fun _ => .pos, rfl⟩).measurable.comp hY |
| have hdom : Integrable (fun p => 2 * ((d : ℝ) * beta + (t : ℝ) + |X p|)) ρ := |
| ((integrable_const _).add hXi).const_mul 2 |
| have hFi : Integrable (finiteIsingBrownianEntropy beta D v B t) ρ := by |
| apply hdom.mono' hF.aestronglyMeasurable |
| exact Eventually.of_forall fun p => by |
| rw [Real.norm_of_nonneg (finiteIsingBrownianEntropy_nonneg beta D v B t p)] |
| exact finiteIsingBrownianEntropy_le_noise hbeta D v B t p |
| refine ⟨hFi, ?_⟩ |
| calc |
| _ ≤ ∫ p, 2 * ((d : ℝ) * beta + (t : ℝ) + |X p|) ∂ρ := |
| integral_mono hFi hdom (finiteIsingBrownianEntropy_le_noise hbeta D v B t) |
| _ = 2 * ((d : ℝ) * beta + (t : ℝ) + ∫ p, |X p| ∂ρ) := by |
| rw [integral_const_mul, integral_add (integrable_const _) hXi] |
| simp [ρ] |
| _ ≤ _ := mul_le_mul_of_nonneg_left |
| (add_le_add le_rfl (integral_abs_gaussian_le_sqrt hX)) (by norm_num) |
| |
| theorem HasFreeIsingLaw.integral_finiteIsingBrownianNormalizedEnergy_le |
| {d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω] |
| {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (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)) |
| (D : Finset (TreeVertex d)) (hD : ConnectedFinset D) (v : D) |
| {T : ℝ} (hT : 0 ≤ T) : |
| let ρ := μ.prod (Measure.infinitePi ν) |
| let E : ℝ → ℝ := fun t => ∫ p, |
| ∑ u : D, finiteIsingBrownianNormalizedCovariance beta D v u B t p ^ 2 ∂ρ |
| IntervalIntegrable E volume 0 T ∧ |
| (∫ t in 0..T, E t) ≤ (∫ p, finiteIsingBrownianEntropy beta D v B T p ∂ρ) ∧ |
| 0 ≤ (∫ t in 0..T, E t) ∧ |
| (∫ t in 0..T, E t) ≤ 2 * ((d : ℝ) * beta + T + Real.sqrt T) := by |
| classical |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| let π := μ.map D.restrict |
| let : IsProbabilityMeasure π := |
| (Measure.isProbabilityMeasure_map_iff D.measurable_restrict.aemeasurable).mpr inferInstance |
| let spin : (D → IsingSpin) → D → ℝ := fun eta u => spinSign (eta u) |
| let q : (D → IsingSpin) → ℝ := fun eta => spinSign (eta v) |
| let F : (D → ℝ) → ℝ := FinitePosterior.fieldEntropy (finiteIsingEnergy beta D) spin q |
| let C : (D → ℝ) → D → ℝ := |
| FinitePosterior.normalizedFieldCovariance (finiteIsingEnergy beta D) spin q |
| let ρ := μ.prod (Measure.infinitePi ν) |
| let ρG := π.prod (Measure.pi fun _ : D => gaussianReal 0 1) |
| let EG : ℝ → ℝ := fun t => ∫ p, |
| ∑ u : D, C (FinitePosterior.finiteStandardObservation spin t p) u ^ 2 ∂ρG |
| let MG : ℝ → ℝ := fun t => ∫ p, |
| F (FinitePosterior.finiteStandardObservation spin t p) ∂ρG |
| let E : ℝ → ℝ := fun t => ∫ p, |
| ∑ u : D, finiteIsingBrownianNormalizedCovariance beta D v u B t p ^ 2 ∂ρ |
| let M : ℝ → ℝ := fun t => ∫ p, finiteIsingBrownianEntropy beta D v B t p ∂ρ |
| have hcoords (eta : D → IsingSpin) (u : D) : spin eta u = -1 ∨ spin eta u = 1 := by |
| dsimp only [spin] |
| cases eta u <;> simp [spinSign] |
| have hneg : ∃ eta, q eta = -1 := ⟨fun _ => .neg, rfl⟩ |
| have hpos : ∃ eta, q eta = 1 := ⟨fun _ => .pos, rfl⟩ |
| have hfm : Measurable F := |
| (FinitePosterior.continuous_fieldEntropy _ _ _ (fun eta => hcoords eta v) hneg hpos).measurable |
| have hcm : Measurable (fun y => ∑ u : D, C y u ^ 2) := |
| (FinitePosterior.continuous_normalizedRowEnergy _ _ _ (fun eta => hcoords eta v) hneg hpos).measurable |
| have hE (t : ℝ) (ht : 0 ≤ t) : E t = EG t := by |
| have h := integral_finiteIsingBrownian_observation_eq_standard μ ν hB hmB D |
| t.toNNReal (fun y => ∑ u : D, C y u ^ 2) hcm |
| unfold E EG finiteIsingBrownianNormalizedCovariance finiteIsingBrownianObservation |
| simpa only [C, spin, q, ρ, ρG, π, Real.coe_toNNReal t ht] using h |
| have hM (t : ℝ) (ht : 0 ≤ t) : M t = MG t := by |
| have h := integral_finiteIsingBrownian_observation_eq_standard μ ν hB hmB D |
| t.toNNReal F hfm |
| unfold M MG finiteIsingBrownianEntropy finiteIsingBrownianObservation |
| simpa only [F, spin, q, ρ, ρG, π, Real.coe_toNNReal t ht] using h |
| have hstd := FinitePosterior.integral_finiteStandardNormalizedEnergy_le π |
| (finiteIsingEnergy beta D) (hμ.restrict_real_eq_normalized_exp D hD) |
| spin q (fun eta => hcoords eta v) hneg hpos hcoords hT |
| change IntervalIntegrable EG volume 0 T ∧ |
| (∫ t in 0..T, EG t) ≤ MG T - F 0 ∧ 0 ≤ (∫ t in 0..T, EG t) at hstd |
| have hzero : F 0 = 0 := finiteIsingEntropy_zero beta D v |
| have hEq : EqOn EG E (uIcc 0 T) := by |
| intro t ht |
| exact (hE t (by simpa only [min_eq_left hT] using ht.1)).symm |
| have hInt : (∫ t in 0..T, EG t) = ∫ t in 0..T, E t := |
| intervalIntegral.integral_congr hEq |
| have hle : (∫ t in 0..T, E t) ≤ M T := by |
| simpa only [hzero, sub_zero, hInt, ← hM T hT] using hstd.2.1 |
| have hterminal : M T ≤ 2 * ((d : ℝ) * beta + T + Real.sqrt T) := by |
| have h := (finiteIsingBrownianEntropy_moment hbeta μ ν hB hmB D v T.toNNReal).2 |
| simpa only [M, ρ, Real.coe_toNNReal T hT] using h |
| change IntervalIntegrable E volume 0 T ∧ (∫ t in 0..T, E t) ≤ M T ∧ |
| 0 ≤ (∫ t in 0..T, E t) ∧ (∫ t in 0..T, E t) ≤ _ |
| exact ⟨hstd.1.congr (fun t ht => hEq (uIoc_subset_uIcc ht)), hle, |
| by simpa only [hInt] using hstd.2.2, hle.trans hterminal⟩ |
| |
| theorem HasFreeIsingLaw.lintegral_finiteIsingBrownianNormalizedEnergy_le |
| {d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω] |
| {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (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)) |
| (D : Finset (TreeVertex d)) (hD : ConnectedFinset D) (v : D) |
| {T : ℝ} (hT : 0 ≤ T) : |
| (∫⁻ t in Ioc 0 T, ∫⁻ p, |
| ENNReal.ofReal (∑ u : D, finiteIsingBrownianNormalizedCovariance beta D v u B t p ^ 2) |
| ∂μ.prod (Measure.infinitePi ν)) ≤ |
| ENNReal.ofReal (2 * ((d : ℝ) * beta + T + Real.sqrt T)) := by |
| classical |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| have (i : TreeVertex d) : IsProbabilityMeasure (ν i) := |
| (hB i).isGaussianProcess.isProbabilityMeasure |
| let ρ := μ.prod (Measure.infinitePi ν) |
| let R := fun t p => ∑ u : D, finiteIsingBrownianNormalizedCovariance beta D v u B t p ^ 2 |
| let E : ℝ → ℝ := fun t => ∫ p, R t p ∂ρ |
| have hRpos (t : ℝ) (p) : 0 ≤ R t p := Finset.sum_nonneg fun u _ => sq_nonneg _ |
| have hRI (t : ℝ) : Integrable (R t) ρ := by |
| have hY : Measurable (finiteIsingBrownianObservation B D t) := by |
| apply Measurable.of_eval |
| intro w |
| exact (((measurable_of_finite spinSign).comp |
| ((measurable_pi_apply (w : TreeVertex d)).comp measurable_fst)).const_mul t).add |
| ((hmB w t.toNNReal).comp ((measurable_pi_apply (w : TreeVertex d)).comp measurable_snd)) |
| have hs (eta : D → IsingSpin) : spinSign (eta v) = -1 ∨ spinSign (eta v) = 1 := by |
| cases eta v <;> simp [spinSign] |
| have hn : ∃ eta : D → IsingSpin, spinSign (eta v) = -1 := ⟨fun _ => .neg, rfl⟩ |
| have hp : ∃ eta : D → IsingSpin, spinSign (eta v) = 1 := ⟨fun _ => .pos, rfl⟩ |
| apply Integrable.of_bound |
| ((FinitePosterior.continuous_normalizedRowEnergy (finiteIsingEnergy beta D) |
| (fun (eta : D → IsingSpin) w => spinSign (eta w)) (fun eta => spinSign (eta v)) |
| hs hn hp).measurable.comp hY).aestronglyMeasurable (Fintype.card D) |
| exact Eventually.of_forall fun p => by |
| simpa only [Real.norm_eq_abs, Function.comp_def] using |
| FinitePosterior.abs_normalizedRowEnergy_le_card (finiteIsingEnergy beta D) |
| (fun (eta : D → IsingSpin) w => spinSign (eta w)) (fun eta => spinSign (eta v)) |
| (finiteIsingBrownianObservation B D t p) hs hn hp |
| (fun eta w => by cases eta w <;> norm_num [spinSign]) |
| have hinner (t : ℝ) : (∫⁻ p, ENNReal.ofReal (R t p) ∂ρ) = ENNReal.ofReal (E t) := |
| (ofReal_integral_eq_lintegral_ofReal (hRI t) (Eventually.of_forall (hRpos t))).symm |
| have hbound := hμ.integral_finiteIsingBrownianNormalizedEnergy_le hbeta ν hB hmB D hD v hT |
| have hEI : Integrable E (volume.restrict (Ioc 0 T)) := |
| (intervalIntegrable_iff_integrableOn_Ioc_of_le hT).mp hbound.1 |
| change (∫⁻ t in Ioc 0 T, ∫⁻ p, ENNReal.ofReal (R t p) ∂ρ) ≤ _ |
| simp_rw [hinner] |
| rw [← ofReal_integral_eq_lintegral_ofReal hEI |
| (Eventually.of_forall fun t => integral_nonneg (hRpos t))] |
| rw [← intervalIntegral.integral_of_le hT] |
| exact ENNReal.ofReal_le_ofReal hbound.2.2.2 |
| |
| end Problem367 |
| |
| end |
| |
| end OAI |
| |
