| import OAI.Probability.FreeIsing.Ising.ObservationResponseEnergy |
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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.IsingObservationResponseEnergy |
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| open IsingObservationInnovation IsingObservationFactor CriticalCanonicalRowEnergy |
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| variable {d : ℕ} {beta : ℝ} |
| variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ] |
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| def timeEnergyConstant (d : ℕ) (beta : ℝ) : ℝ := |
| 1 + (d : ℝ) * Real.tanh beta ^ 2 / |
| ComponentSignalAttenuation.lossCoefficient (Real.tanh beta) |
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| theorem row_bound_le_time_majorant |
| (hb : 0 ≤ beta) (hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1) |
| {t : ℝ} (ht : 0 < t) : |
| 1 + (d : ℝ) * Real.tanh beta ^ 2 / rate beta t.toNNReal ≤ |
| timeEnergyConstant d beta * (1 + t ^ (-(1 : ℝ) / 2)) := by |
| have hC : 0 < ComponentSignalAttenuation.lossCoefficient (Real.tanh beta) := by |
| simpa [rate] using (rate_pos hb hc (by norm_num : (0 : ℝ≥0) < 1)) |
| have hK : 0 ≤ (d : ℝ) * Real.tanh beta ^ 2 / |
| ComponentSignalAttenuation.lossCoefficient (Real.tanh beta) := by positivity |
| have hs : 0 < Real.sqrt t := Real.sqrt_pos.mpr ht |
| have hp : t ^ (-(1 : ℝ) / 2) = (Real.sqrt t)⁻¹ := by |
| rw [show -(1 : ℝ) / 2 = -((1 : ℝ) / 2) by ring, Real.rpow_neg ht.le, |
| ← Real.sqrt_eq_rpow] |
| rw [hp] |
| unfold rate timeEnergyConstant |
| rw [Real.coe_toNNReal t ht.le] |
| by_cases hs1 : Real.sqrt t ≤ 1 |
| · rw [min_eq_left hs1, div_mul_eq_div_mul_one_div, one_div] |
| nlinarith [inv_nonneg.mpr hs.le] |
| · rw [min_eq_right (le_of_not_ge hs1), mul_one] |
| nlinarith [inv_nonneg.mpr hs.le] |
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| theorem integrable_time_majorant (d : ℕ) (beta : ℝ) {T : ℝ} (hT : 0 ≤ T) : |
| Integrable (fun t : ℝ => timeEnergyConstant d beta * (1 + t ^ (-(1 : ℝ) / 2))) |
| (volume.restrict (Ioc 0 T)) := by |
| have hp : IntervalIntegrable (fun t : ℝ => t ^ (-(1 : ℝ) / 2)) volume 0 T := |
| intervalIntegral.intervalIntegrable_rpow' (by norm_num) |
| exact ((integrable_const (1 : ℝ)).add |
| ((intervalIntegrable_iff_integrableOn_Ioc_of_le hT).mp hp)).const_mul _ |
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| theorem integrable_response_row_time |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d) |
| (hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1) |
| (u : TreeVertex d) {T : ℝ} (hT : 0 ≤ T) : |
| Integrable (fun p : ℝ × Paths d => ∑' v, response beta u v p.1.toNNReal p.2 ^ 2) |
| ((volume.restrict (Ioc 0 T)).prod (observationLaw μ)) := by |
| have hm : Measurable (fun p : ℝ × Paths d => ∑' v, |
| response beta u v p.1.toNNReal p.2 ^ 2) := by |
| apply StronglyMeasurable.measurable |
| apply StronglyMeasurable.tsum |
| intro v |
| exact (((measurable_response_joint beta u v).comp |
| (measurable_real_toNNReal.prodMap measurable_id)).pow_const 2).stronglyMeasurable |
| apply (integrable_prod_iff hm.aestronglyMeasurable).mpr |
| constructor |
| · filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht |
| exact (response_row_energy hμ hb hd hc (Real.toNNReal_pos.mpr ht.1) u).2.1 |
| · apply (integrable_time_majorant d beta hT).mono' |
| hm.aestronglyMeasurable.norm.integral_prod_right' |
| filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht |
| have hn (x : Paths d) : 0 ≤ ∑' v, response beta u v t.toNNReal x ^ 2 := |
| tsum_nonneg (fun v => sq_nonneg _) |
| simp_rw [Real.norm_eq_abs, abs_of_nonneg (hn _)] |
| rw [abs_of_nonneg (integral_nonneg hn)] |
| exact (response_row_energy hμ hb hd hc (Real.toNNReal_pos.mpr ht.1) u).2.2.trans |
| (row_bound_le_time_majorant hb hc ht.1) |
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| theorem integrable_copy_response_row_time |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d) |
| (hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1) |
| (u : TreeVertex d) {T : ℝ} (hT : 0 ≤ T) (j : Bool) : |
| Integrable (fun p : ℝ × (Paths d × (Paths d × Paths d)) => ∑' v, |
| response beta u v p.1.toNNReal (copyObservation j p.2) ^ 2) |
| ((volume.restrict (Ioc 0 T)).prod (copies beta μ)) := |
| ((MeasurePreserving.id (volume.restrict (Ioc 0 T))).prod |
| (measurePreserving_copyObservation (beta := beta) (μ := μ) j)).integrable_comp_of_integrable |
| (integrable_response_row_time hμ hb hd hc u hT) |
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| theorem integral_time_majorant (d : ℕ) (beta : ℝ) {T : ℝ} (hT : 0 ≤ T) : |
| (∫ t in Ioc 0 T, timeEnergyConstant d beta * (1 + t ^ (-(1 : ℝ) / 2))) = |
| timeEnergyConstant d beta * (T + 2 * Real.sqrt T) := by |
| have hp : IntervalIntegrable (fun t : ℝ => t ^ (-(1 : ℝ) / 2)) volume 0 T := |
| intervalIntegral.intervalIntegrable_rpow' (by norm_num) |
| rw [integral_const_mul] |
| congr 1 |
| rw [← intervalIntegral.integral_of_le hT, |
| intervalIntegral.integral_add intervalIntegrable_const hp, |
| intervalIntegral.integral_const, integral_rpow (Or.inl (by norm_num))] |
| norm_num |
| rw [← Real.sqrt_eq_rpow] |
| ring |
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| theorem lintegral_response_row_time_le |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d) |
| (hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1) |
| (u : TreeVertex d) {T : ℝ} (hT : 0 ≤ T) : |
| (∫⁻ p : ℝ × Paths d, ∑' v, ENNReal.ofReal |
| (response beta u v p.1.toNNReal p.2 ^ 2) |
| ∂(volume.restrict (Ioc 0 T)).prod (observationLaw μ)) ≤ |
| ENNReal.ofReal (timeEnergyConstant d beta * (T + 2 * Real.sqrt T)) := by |
| have hm : Measurable (fun p : ℝ × Paths d => ∑' v, ENNReal.ofReal |
| (response beta u v p.1.toNNReal p.2 ^ 2)) := |
| Measurable.tsum fun v => (((measurable_response_joint beta u v).comp |
| (measurable_real_toNNReal.prodMap measurable_id)).pow_const 2).ennreal_ofReal |
| have hC : 0 < ComponentSignalAttenuation.lossCoefficient (Real.tanh beta) := by |
| simpa [rate] using (rate_pos hb hc (by norm_num : (0 : ℝ≥0) < 1)) |
| have hK : 0 ≤ timeEnergyConstant d beta := by |
| unfold timeEnergyConstant |
| positivity |
| calc |
| _ = ∫⁻ t in Ioc 0 T, ∫⁻ x, ∑' v, ENNReal.ofReal |
| (response beta u v t.toNNReal x ^ 2) ∂observationLaw μ := |
| lintegral_prod _ hm.aemeasurable |
| _ ≤ ∫⁻ t in Ioc 0 T, |
| ENNReal.ofReal (timeEnergyConstant d beta * (1 + t ^ (-(1 : ℝ) / 2))) := by |
| apply lintegral_mono_ae |
| filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht |
| exact (lintegral_response_row_le hμ hb hd hc (Real.toNNReal_pos.mpr ht.1) u).trans |
| (ENNReal.ofReal_le_ofReal (row_bound_le_time_majorant hb hc ht.1)) |
| _ = ENNReal.ofReal (∫ t in Ioc 0 T, |
| timeEnergyConstant d beta * (1 + t ^ (-(1 : ℝ) / 2))) := by |
| symm |
| apply ofReal_integral_eq_lintegral_ofReal (integrable_time_majorant d beta hT) |
| filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht |
| exact mul_nonneg hK (add_nonneg zero_le_one (Real.rpow_nonneg ht.1.le _)) |
| _ = _ := congrArg ENNReal.ofReal (integral_time_majorant d beta hT) |
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| theorem lintegral_copy_response_row_time_le |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d) |
| (hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1) |
| (u : TreeVertex d) {T : ℝ} (hT : 0 ≤ T) (j : Bool) : |
| (∫⁻ p : ℝ × (Paths d × (Paths d × Paths d)), ∑' v, ENNReal.ofReal |
| (response beta u v p.1.toNNReal (copyObservation j p.2) ^ 2) |
| ∂(volume.restrict (Ioc 0 T)).prod (copies beta μ)) ≤ |
| ENNReal.ofReal (timeEnergyConstant d beta * (T + 2 * Real.sqrt T)) := by |
| have hp := (MeasurePreserving.id (volume.restrict (Ioc 0 T))).prod |
| (measurePreserving_copyObservation (beta := beta) (μ := μ) j) |
| have hm : Measurable (fun p : ℝ × Paths d => ∑' v, ENNReal.ofReal |
| (response beta u v p.1.toNNReal p.2 ^ 2)) := |
| Measurable.tsum fun v => (((measurable_response_joint beta u v).comp |
| (measurable_real_toNNReal.prodMap measurable_id)).pow_const 2).ennreal_ofReal |
| exact (hp.hasLaw.lintegral_comp hm.aemeasurable).trans_le |
| (lintegral_response_row_time_le hμ hb hd hc u hT) |
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| theorem tendsto_copy_response_row_time_zero |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d) |
| (hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1) |
| (u : TreeVertex d) (j : Bool) : |
| Tendsto (fun T : ℝ≥0 => ∫⁻ p : ℝ × (Paths d × (Paths d × Paths d)), |
| ∑' v, ENNReal.ofReal (response beta u v p.1.toNNReal (copyObservation j p.2) ^ 2) |
| ∂(volume.restrict (Ioc 0 (T : ℝ))).prod (copies beta μ)) (𝓝 0) (𝓝 0) := by |
| have hcont : Continuous (fun T : ℝ≥0 => ENNReal.ofReal |
| (timeEnergyConstant d beta * ((T : ℝ) + 2 * Real.sqrt (T : ℝ)))) := |
| ENNReal.continuous_ofReal.comp |
| (continuous_const.mul (NNReal.continuous_coe.add |
| (continuous_const.mul (Real.continuous_sqrt.comp NNReal.continuous_coe)))) |
| have hlim : Tendsto (fun T : ℝ≥0 => ENNReal.ofReal |
| (timeEnergyConstant d beta * ((T : ℝ) + 2 * Real.sqrt (T : ℝ)))) (𝓝 0) (𝓝 0) := by |
| simpa only [NNReal.coe_zero, Real.sqrt_zero, mul_zero, add_zero, ENNReal.ofReal_zero] |
| using hcont.tendsto 0 |
| exact tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hlim |
| (Eventually.of_forall fun _ => bot_le) |
| (Eventually.of_forall fun T => lintegral_copy_response_row_time_le hμ hb hd hc u T.property j) |
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| end Problem367.IsingObservationResponseEnergy |
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| end |
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| end OAI |
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