openAI-math-dataset / lean /OAI /Probability /FreeIsing /Energy /NormalizedCriticalInitialEnergy.lean
| import OAI.Probability.FreeIsing.Conditioning.FiniteEntropySmallTime | |
| import OAI.Probability.FreeIsing.Stochastic.NormalizedBrownianVarianceComparison | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set Filter | |
| open scoped NNReal ENNReal Topology | |
| namespace Problem367.NormalizedCriticalInitialEnergy | |
| variable {d : ℕ} {Ω : Type*} [MeasurableSpace Ω] | |
| def informationConstant (beta : ℝ) : ℝ := | |
| (1 + Real.tanh beta ^ 2) * | |
| Real.sqrt (1 / ((1 - Real.tanh beta ^ 2) * Real.tanh beta ^ 4)) | |
| lemma informationConstant_nonneg (beta : ℝ) : 0 ≤ informationConstant beta := by | |
| unfold informationConstant | |
| positivity | |
| def energyConstant (d : ℕ) (beta : ℝ) : ℝ := | |
| informationConstant beta / GaussianVarianceLower.varianceLower 1 (1 + (d : ℝ) * beta) | |
| lemma energyConstant_nonneg (d : ℕ) (beta : ℝ) : 0 ≤ energyConstant d beta := | |
| div_nonneg (informationConstant_nonneg beta) (GaussianVarianceLower.varianceLower_pos _ _).le | |
| theorem lintegral_finite_energy_le | |
| (hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} | |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) | |
| (hcrit : ((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)) | |
| (D : Finset (TreeVertex d)) (hD : ConnectedFinset D) (v : D) | |
| (hv : v.1.1 = []) (T : ℝ≥0) (hT : (T : ℝ) ≤ 1) : | |
| (∫⁻ t in Ioc 0 (T : ℝ), ∫⁻ p, | |
| ENNReal.ofReal (∑ w : D, finiteIsingBrownianNormalizedCovariance beta D v w B t p ^ 2) | |
| ∂μ.prod (Measure.infinitePi ν)) ≤ | |
| ENNReal.ofReal (energyConstant d beta * Real.sqrt (T : ℝ)) := by | |
| classical | |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure | |
| have (i : TreeVertex d) : IsProbabilityMeasure (ν i) := | |
| (hB i).isGaussianProcess.isProbabilityMeasure | |
| have h := NormalizedBrownianVarianceComparison.lintegral_energy_le_information | |
| hμ hb ν hB hmB D hD v hv (by norm_num : (0 : ℝ) ≤ 1) T.property hT | |
| refine h.trans (ENNReal.ofReal_le_ofReal ?_) | |
| have hm := div_le_div_of_nonneg_right | |
| (hμ.finiteIsingBrownian_critical_information_upper hd hb hcrit ν hB hmB D hD v T hT) | |
| (GaussianVarianceLower.varianceLower_pos 1 (1 + (d : ℝ) * beta)).le | |
| change _ ≤ informationConstant beta / GaussianVarianceLower.varianceLower 1 | |
| (1 + (d : ℝ) * beta) * Real.sqrt (T : ℝ) | |
| convert hm using 1 <;> first | rfl | (dsimp [informationConstant]; ring) | |
| theorem lintegral_infinite_energy_le | |
| (hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} | |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) | |
| (hcrit : ((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) (hv : v.1 = []) (T : ℝ≥0) (hT : (T : ℝ) ≤ 1) : | |
| (∫⁻ q, ∑' w, ENNReal.ofReal | |
| (NormalizedInfiniteTimeEnergy.infiniteRow beta v B w q ^ 2) | |
| ∂(volume.restrict (Ioc 0 (T : ℝ))).prod (μ.prod (Measure.infinitePi ν))) ≤ | |
| ENNReal.ofReal (energyConstant d beta * Real.sqrt (T : ℝ)) := by | |
| classical | |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure | |
| have (i : TreeVertex d) : IsProbabilityMeasure (ν i) := | |
| (hB i).isGaussianProcess.isProbabilityMeasure | |
| apply NormalizedInfiniteTimeEnergy.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 (∑ w : D, | |
| finiteIsingBrownianNormalizedCovariance beta D r w B q.1 q.2 ^ 2) | |
| have hR : Measurable R := by | |
| have heq : R = fun q => ∑' w, ENNReal.ofReal | |
| (NormalizedInfiniteTimeEnergy.finiteRow beta D v B w q ^ 2) := by | |
| funext q | |
| exact (NormalizedInfiniteTimeEnergy.tsum_finiteRow_sq beta D r B q).symm | |
| rw [heq] | |
| exact Measurable.tsum (fun w => | |
| ((NormalizedInfiniteTimeEnergy.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 | |
| _ ≤ _ := lintegral_finite_energy_le hd hμ hb hcrit ν hB hmB | |
| D (connectedExhaustion_connected v n) r hv T hT | |
| theorem tendsto_infinite_energy_zero | |
| (hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} | |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) | |
| (hcrit : ((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) (hv : v.1 = []) : | |
| Tendsto (fun T : ℝ≥0 => ∫⁻ q, ∑' w, ENNReal.ofReal | |
| (NormalizedInfiniteTimeEnergy.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 (energyConstant d beta * Real.sqrt (T : ℝ))) := | |
| ENNReal.continuous_ofReal.comp | |
| (continuous_const.mul (Real.continuous_sqrt.comp NNReal.continuous_coe)) | |
| have hlim : Tendsto (fun T : ℝ≥0 => | |
| ENNReal.ofReal (energyConstant d beta * Real.sqrt (T : ℝ))) (𝓝 0) (𝓝 0) := by | |
| simpa only [NNReal.coe_zero, Real.sqrt_zero, mul_zero, ENNReal.ofReal_zero] using hc.tendsto 0 | |
| apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hlim | |
| (Eventually.of_forall fun _ => bot_le) | |
| filter_upwards [eventually_lt_nhds (by norm_num : (0 : ℝ≥0) < 1)] with T hT | |
| exact lintegral_infinite_energy_le hd hμ hb hcrit ν hB hmB hjoint v hv T | |
| (by exact_mod_cast hT.le) | |
| end Problem367.NormalizedCriticalInitialEnergy | |
| end | |
| end OAI | |