| import OAI.Probability.FreeIsing.Conditioning.FiniteEntropySmallTime |
| import OAI.Probability.FreeIsing.Energy.NormalizedInfiniteTimeEnergy |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped NNReal ENNReal Topology |
|
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| namespace Problem367 |
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| local instance : Nonempty IsingSpin := ⟨.neg⟩ |
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|
| theorem HasFreeIsingLaw.lintegral_finiteIsingBrownianNormalizedEnergy_le_entropy |
| {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 (∫ p, finiteIsingBrownianEntropy beta D v B T p |
| ∂μ.prod (Measure.infinitePi ν)) := 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.2.1 |
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| namespace NormalizedInfiniteTimeEnergy |
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|
| theorem lintegral_infiniteRow_le_criticalEnvelope |
| {d : ℕ} (hd : 3 ≤ d) {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω] |
| {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcritical : ((d - 1 : ℕ) : ℝ) * (Real.tanh beta) ^ 2 = 1) |
| (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)) |
| (hjoint : ∀ i, Measurable (fun z : ℝ≥0 × Ω => B i z.1 z.2)) |
| (v : TreeVertex d) (T : ℝ≥0) (hT : (T : ℝ) ≤ 1) : |
| (∫⁻ q, ∑' w, ENNReal.ofReal (infiniteRow beta v B w q ^ 2) |
| ∂(volume.restrict (Ioc 0 (T : ℝ))).prod (μ.prod (Measure.infinitePi ν))) ≤ |
| ENNReal.ofReal (criticalEntropyEnvelope d beta T) := by |
| let := hμ.isProbabilityMeasure |
| have (i : TreeVertex d) : IsProbabilityMeasure (ν i) := |
| (hB i).isGaussianProcess.isProbabilityMeasure |
| apply lintegral_infiniteRow_le_of_finite hμ hb ν hB hmB hjoint |
| (connectedExhaustion v) (connectedExhaustion_monotone v) |
| (connectedExhaustion_connected v) |
| (fun u => (eventually_mem_connectedExhaustion v u).exists) |
| v (mem_connectedExhaustion v) T |
| intro n |
| let D := connectedExhaustion v n |
| let r : D := ⟨v, mem_connectedExhaustion v n⟩ |
| let R : ℝ × ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) → ℝ≥0∞ := |
| fun q => ENNReal.ofReal (∑ u : D, |
| finiteIsingBrownianNormalizedCovariance beta D r u B q.1 q.2 ^ 2) |
| have hR : Measurable R := by |
| have heq : R = fun q => ∑' w, ENNReal.ofReal (finiteRow beta D v B w q ^ 2) := by |
| funext q |
| exact (tsum_finiteRow_sq beta D r B q).symm |
| rw [heq] |
| exact Measurable.tsum (fun w => |
| ((measurable_finiteRow beta D v hjoint w).pow_const 2).ennreal_ofReal) |
| calc |
| _ = ∫⁻ q, R q |
| ∂(volume.restrict (Ioc 0 (T : ℝ))).prod (μ.prod (Measure.infinitePi ν)) := by |
| apply lintegral_congr |
| intro q |
| exact (ENNReal.ofReal_sum_of_nonneg (fun _ _ => sq_nonneg _)).symm |
| _ = ∫⁻ t in Ioc 0 (T : ℝ), ∫⁻ p, R (t, p) ∂μ.prod (Measure.infinitePi ν) := |
| lintegral_prod R hR.aemeasurable |
| _ ≤ ENNReal.ofReal (∫ p, finiteIsingBrownianEntropy beta D r B T p |
| ∂μ.prod (Measure.infinitePi ν)) := |
| hμ.lintegral_finiteIsingBrownianNormalizedEnergy_le_entropy hb ν hB hmB |
| D (connectedExhaustion_connected v n) r T.property |
| _ ≤ _ := ENNReal.ofReal_le_ofReal |
| (hμ.finiteIsingBrownianEntropy_le_criticalEnvelope hd hb hcritical ν hB hmB |
| D (connectedExhaustion_connected v n) r T hT) |
|
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| theorem tendsto_lintegral_infiniteRow_critical_zero |
| {d : ℕ} (hd : 3 ≤ d) {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω] |
| {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcritical : ((d - 1 : ℕ) : ℝ) * (Real.tanh beta) ^ 2 = 1) |
| (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)) |
| (hjoint : ∀ i, Measurable (fun z : ℝ≥0 × Ω => B i z.1 z.2)) |
| (v : TreeVertex d) : |
| Tendsto (fun T : ℝ≥0 => |
| ∫⁻ q, ∑' w, ENNReal.ofReal (infiniteRow beta v B w q ^ 2) |
| ∂(volume.restrict (Ioc 0 (T : ℝ))).prod (μ.prod (Measure.infinitePi ν))) |
| (𝓝 0) (𝓝 0) := by |
| have hc : Continuous (fun T : ℝ≥0 => |
| ENNReal.ofReal (criticalEntropyEnvelope d beta T)) := |
| ENNReal.continuous_ofReal.comp ((continuous_criticalEntropyEnvelope d beta).comp |
| NNReal.continuous_coe) |
| have hlim : Tendsto (fun T : ℝ≥0 => ENNReal.ofReal (criticalEntropyEnvelope d beta T)) |
| (𝓝 0) (𝓝 0) := by simpa using hc.tendsto 0 |
| apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hlim |
| · exact Eventually.of_forall fun _ => zero_le |
| · filter_upwards [eventually_lt_nhds (by norm_num : (0 : ℝ≥0) < 1)] with T hT |
| exact lintegral_infiniteRow_le_criticalEnvelope hd hμ hb hcritical ν hB hmB hjoint |
| v T (by exact_mod_cast hT.le) |
|
|
| end NormalizedInfiniteTimeEnergy |
| end Problem367 |
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| end |
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| end OAI |
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