| import OAI.Probability.FreeIsing.Conditioning.CovarianceLimit |
| import OAI.Probability.FreeIsing.Posterior.ProgressiveIsingPosterior |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory Filter |
| open scoped Topology NNReal |
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| namespace Problem367 |
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| private theorem measurable_artanh_field : Measurable Real.artanh := by |
| unfold Real.artanh |
| fun_prop |
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| theorem tendsto_integral_sq_sub_posterior_fields |
| {Ω : Type*} [m : MeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ] |
| (F : Filtration ℕ m) (f X : Ω → ℝ) {K : ℝ} |
| (hb : ∀ n, ∀ᵐ ω ∂μ, |
| Real.tanh (X ω - K) ≤ (μ[f | F n]) ω ∧ |
| (μ[f | F n]) ω ≤ Real.tanh (X ω + K)) : |
| Tendsto (fun n => ∫ ω, |
| (Real.artanh ((μ[f | F n]) ω) - |
| Real.artanh ((μ[f | ⨆ n, F n]) ω)) ^ 2 ∂μ) atTop (𝓝 0) := by |
| have hs : (⨆ n, F n) ≤ m := iSup_le (fun n => F.le n) |
| have hm (n : ℕ) : AEStronglyMeasurable |
| (fun ω => Real.artanh ((μ[f | F n]) ω)) μ := |
| (measurable_artanh_field.comp |
| (stronglyMeasurable_condExp.mono (F.le n)).measurable).aestronglyMeasurable |
| have hm' : AEStronglyMeasurable |
| (fun ω => Real.artanh ((μ[f | ⨆ n, F n]) ω)) μ := |
| (measurable_artanh_field.comp |
| (stronglyMeasurable_condExp.mono hs).measurable).aestronglyMeasurable |
| have hlim := tendsto_posterior_fields_ae μ F f X K hb |
| have hbfield (n : ℕ) : ∀ᵐ ω ∂μ, |
| |Real.artanh ((μ[f | F n]) ω) - X ω| ≤ K := by |
| filter_upwards [hb n] with ω hω |
| exact (posterior_field_limit_of_tanh_bounds |
| (u := fun _ : ℕ => (μ[f | F n]) ω) tendsto_const_nhds |
| (fun _ => hω)).2.1 |
| have hbound (n : ℕ) : ∀ᵐ ω ∂μ, |
| ‖(Real.artanh ((μ[f | F n]) ω) - |
| Real.artanh ((μ[f | ⨆ n, F n]) ω)) ^ 2‖ ≤ (2 * K) ^ 2 := by |
| filter_upwards [hbfield n, hlim] with ω hn hω |
| have hd : |Real.artanh ((μ[f | F n]) ω) - |
| Real.artanh ((μ[f | ⨆ n, F n]) ω)| ≤ 2 * K := by |
| calc |
| _ = |(Real.artanh ((μ[f | F n]) ω) - X ω) - |
| (Real.artanh ((μ[f | ⨆ n, F n]) ω) - X ω)| := by congr 1; ring |
| _ ≤ |Real.artanh ((μ[f | F n]) ω) - X ω| + |
| |Real.artanh ((μ[f | ⨆ n, F n]) ω) - X ω| := abs_sub _ _ |
| _ ≤ 2 * K := by linarith [hω.2.1] |
| rw [Real.norm_of_nonneg (sq_nonneg _)] |
| simpa only [sq_abs] using pow_le_pow_left₀ (abs_nonneg _) hd 2 |
| have hzero : ∀ᵐ ω ∂μ, Tendsto |
| (fun n => (Real.artanh ((μ[f | F n]) ω) - |
| Real.artanh ((μ[f | ⨆ n, F n]) ω)) ^ 2) atTop (𝓝 (0 : ℝ)) := by |
| filter_upwards [hlim] with ω hω |
| simpa using (hω.2.2.sub_const (Real.artanh ((μ[f | ⨆ n, F n]) ω))).pow 2 |
| simpa using tendsto_integral_of_dominated_convergence |
| (fun _ : Ω => (2 * K) ^ 2) (fun n => ((hm n).sub hm').pow 2) |
| (integrable_const _) hbound hzero |
|
|
| open PlantedSymmetry |
|
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| theorem HasFreeIsingLaw.finiteGibbsField_tendsto_ae |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (t : ℝ≥0) (v : TreeVertex d) : |
| ∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t), |
| Tendsto (fun n => Real.artanh (finiteGibbsSpinMean beta v n (observation t p))) |
| atTop (𝓝 (Real.artanh (infiniteGibbsSpinMean beta v (observation t p)))) := by |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| have hb (n : ℕ) := hμ.finiteObservation_spin_tanh_bounds hbeta |
| (connectedExhaustion v n) (connectedExhaustion_connected v n) t |
| ⟨v, mem_connectedExhaustion v n⟩ |
| have hlim := tendsto_posterior_fields_ae |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) |
| (finiteObservationFiltration (connectedExhaustion v) |
| (connectedExhaustion_monotone v) t) |
| (fun p => spinSign (p.1 v)) (fun p => observation t p v) ((d : ℝ) * beta) hb |
| rw [finiteObservationFiltration_iSup (connectedExhaustion v) |
| (connectedExhaustion_monotone v) |
| (fun w => (eventually_mem_connectedExhaustion v w).exists) t] at hlim |
| have hce (n : ℕ) := hμ.finiteObservation_condExp (connectedExhaustion v n) |
| (connectedExhaustion_connected v n) t |
| (fun eta => spinSign (eta ⟨v, mem_connectedExhaustion v n⟩)) |
| filter_upwards [hlim, ae_all_iff.mpr hce, |
| hμ.infiniteGibbsSpinMean_eq_posterior t v] with p hp hce hinf |
| convert hp.2.2 using 1 |
| · ext n |
| exact congrArg Real.artanh (hce n).symm |
| · exact congrArg (fun x : ℝ => 𝓝 (Real.artanh x)) hinf |
|
|
| theorem HasFreeIsingLaw.finiteGibbsField_tendsto_meanSquare |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (t : ℝ≥0) (v : TreeVertex d) : |
| Tendsto (fun n => ∫ p, |
| (Real.artanh (finiteGibbsSpinMean beta v n (observation t p)) - |
| Real.artanh (infiniteGibbsSpinMean beta v (observation t p))) ^ 2 |
| ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) atTop (𝓝 0) := by |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| have hb (n : ℕ) := hμ.finiteObservation_spin_tanh_bounds hbeta |
| (connectedExhaustion v n) (connectedExhaustion_connected v n) t |
| ⟨v, mem_connectedExhaustion v n⟩ |
| have hlim := tendsto_integral_sq_sub_posterior_fields |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) |
| (finiteObservationFiltration (connectedExhaustion v) |
| (connectedExhaustion_monotone v) t) |
| (fun p => spinSign (p.1 v)) (fun p => observation t p v) |
| hb |
| rw [finiteObservationFiltration_iSup (connectedExhaustion v) |
| (connectedExhaustion_monotone v) |
| (fun w => (eventually_mem_connectedExhaustion v w).exists) t] at hlim |
| convert hlim using 1 |
| ext n |
| apply integral_congr_ae |
| filter_upwards [hμ.finiteObservation_condExp (connectedExhaustion v n) |
| (connectedExhaustion_connected v n) t |
| (fun eta => spinSign (eta ⟨v, mem_connectedExhaustion v n⟩)), |
| hμ.infiniteGibbsSpinMean_eq_posterior t v] with p hfinite hinf |
| change _ = (Real.artanh (_) - Real.artanh |
| (posteriorMagnetization (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) t v p)) ^ 2 |
| rw [hinf] |
| congr 2 |
| exact congrArg Real.artanh hfinite.symm |
|
|
| theorem HasFreeIsingLaw.finiteGibbsField_brownian_tendsto_ae |
| {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)) (t : ℝ≥0) (v : TreeVertex d) : |
| ∀ᵐ p ∂μ.prod (Measure.infinitePi ν), |
| let y := fun i => (t : ℝ) * spinSign (p.1 i) + B i t (p.2 i) |
| Tendsto (fun n => Real.artanh (finiteGibbsSpinMean beta v n y)) |
| atTop (𝓝 (Real.artanh (infiniteGibbsSpinMean beta v y))) := by |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| have (i : TreeVertex d) : IsProbabilityMeasure (ν i) := |
| (hB i).isGaussianProcess.isProbabilityMeasure |
| let T : ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) → |
| (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) := |
| Prod.map id (fun omega i => B i t (omega i)) |
| have hT : MeasurePreserving T (μ.prod (Measure.infinitePi ν)) |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) := |
| (MeasurePreserving.id μ).prod (brownian_array_eval_measurePreserving ν hB hmB t) |
| have hlim := hμ.finiteGibbsField_tendsto_ae hbeta t v |
| rw [← hT.map_eq] at hlim |
| have hlim' := ae_of_ae_map hT.measurable.aemeasurable hlim |
| exact hlim' |
|
|
| theorem HasFreeIsingLaw.finiteGibbsField_brownian_tendsto_meanSquare |
| {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)) (t : ℝ≥0) (v : TreeVertex d) : |
| Tendsto (fun n => ∫ p, |
| let y := fun i => (t : ℝ) * spinSign (p.1 i) + B i t (p.2 i) |
| (Real.artanh (finiteGibbsSpinMean beta v n y) - |
| Real.artanh (infiniteGibbsSpinMean beta v y)) ^ 2 |
| ∂μ.prod (Measure.infinitePi ν)) atTop (𝓝 0) := by |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| have (i : TreeVertex d) : IsProbabilityMeasure (ν i) := |
| (hB i).isGaussianProcess.isProbabilityMeasure |
| let T : ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) → |
| (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) := |
| Prod.map id (fun omega i => B i t (omega i)) |
| have hT : MeasurePreserving T (μ.prod (Measure.infinitePi ν)) |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) := |
| (MeasurePreserving.id μ).prod (brownian_array_eval_measurePreserving ν hB hmB t) |
| have hlim := hμ.finiteGibbsField_tendsto_meanSquare hbeta t v |
| convert hlim using 1 |
| ext n |
| let f : ((TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)) → ℝ := fun p => |
| (Real.artanh (finiteGibbsSpinMean beta v n (observation t p)) - |
| Real.artanh (infiniteGibbsSpinMean beta v (observation t p))) ^ 2 |
| have hf : Measurable f := |
| ((measurable_artanh_field.comp ((continuous_finiteGibbsSpinMean beta v n).measurable.comp |
| (measurable_observation t))).sub |
| (measurable_artanh_field.comp ((measurable_infiniteGibbsSpinMean beta v).comp |
| (measurable_observation t)))).pow_const 2 |
| change (∫ p, f (T p) ∂μ.prod (Measure.infinitePi ν)) = |
| ∫ p, f p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t) |
| rw [← hT.map_eq, integral_map hT.measurable.aemeasurable hf.aestronglyMeasurable] |
|
|
| theorem HasFreeIsingLaw.finiteGibbsField_brownian_discrepancy_bound |
| {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)) (t : ℝ≥0) (v : TreeVertex d) (n : ℕ) : |
| ∀ᵐ p ∂μ.prod (Measure.infinitePi ν), |
| let y := fun i => (t : ℝ) * spinSign (p.1 i) + B i t (p.2 i) |
| |Real.artanh (finiteGibbsSpinMean beta v n y) - |
| Real.artanh (infiniteGibbsSpinMean beta v y)| ≤ 2 * ((d : ℝ) * beta) := by |
| filter_upwards [hμ.infiniteGibbsSpinMean_brownian_field_bound hbeta ν hB hmB t v] |
| with p hp |
| let y := fun i => (t : ℝ) * spinSign (p.1 i) + B i t (p.2 i) |
| have hfin := finite_tree_gibbs_field_uniform_bound (connectedExhaustion v n) hbeta |
| ((connectedExhaustion v n).restrict y) ⟨v, mem_connectedExhaustion v n⟩ |
| change |Real.artanh (finiteGibbsSpinMean beta v n y) - y v| ≤ (d : ℝ) * beta at hfin |
| change |Real.artanh (finiteGibbsSpinMean beta v n y) - |
| Real.artanh (infiniteGibbsSpinMean beta v y)| ≤ _ |
| have hinf : |Real.artanh (infiniteGibbsSpinMean beta v y) - y v| ≤ (d : ℝ) * beta := hp.2 |
| calc |
| _ = |(Real.artanh (finiteGibbsSpinMean beta v n y) - y v) - |
| (Real.artanh (infiniteGibbsSpinMean beta v y) - y v)| := by congr 1; ring |
| _ ≤ |Real.artanh (finiteGibbsSpinMean beta v n y) - y v| + |
| |Real.artanh (infiniteGibbsSpinMean beta v y) - y v| := abs_sub _ _ |
| _ ≤ _ := by linarith |
|
|
| theorem HasFreeIsingLaw.finiteGibbsField_brownian_tendsto_timeMeanSquare |
| {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)) |
| (hcB : ∀ i omega, Continuous (fun t => B i t omega)) |
| (τ : Measure ℝ≥0) [IsFiniteMeasure τ] (v : TreeVertex d) : |
| Tendsto (fun n => ∫ q, |
| let y := fun i => (q.1 : ℝ) * spinSign (q.2.1 i) + B i q.1 (q.2.2 i) |
| (Real.artanh (finiteGibbsSpinMean beta v n y) - |
| Real.artanh (infiniteGibbsSpinMean beta v y)) ^ 2 |
| ∂τ.prod (μ.prod (Measure.infinitePi ν))) atTop (𝓝 0) := by |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| have (i : TreeVertex d) : IsProbabilityMeasure (ν i) := |
| (hB i).isGaussianProcess.isProbabilityMeasure |
| let Y : ℝ≥0 × ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) → |
| TreeVertex d → ℝ := fun q i => |
| (q.1 : ℝ) * spinSign (q.2.1 i) + B i q.1 (q.2.2 i) |
| have hY : Measurable Y := by |
| apply Measurable.of_eval |
| intro i |
| have hnoise : Measurable (Function.uncurry (B i)) := |
| measurable_uncurry_of_continuous_of_measurable (hcB i) (hmB i) |
| have hspin : Measurable (fun q : ℝ≥0 × |
| ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) => spinSign (q.2.1 i)) := |
| (measurable_of_finite spinSign).comp |
| ((measurable_pi_apply i).comp (measurable_fst.comp measurable_snd)) |
| exact ((measurable_subtype_coe.comp measurable_fst).mul hspin).add |
| (hnoise.comp (measurable_fst.prodMk |
| ((measurable_pi_apply i).comp (measurable_snd.comp measurable_snd)))) |
| let Z n q := Real.artanh (finiteGibbsSpinMean beta v n (Y q)) - |
| Real.artanh (infiniteGibbsSpinMean beta v (Y q)) |
| have hZ (n : ℕ) : Measurable (Z n) := |
| (measurable_artanh_field.comp ((continuous_finiteGibbsSpinMean beta v n).measurable.comp hY)).sub |
| (measurable_artanh_field.comp ((measurable_infiniteGibbsSpinMean beta v).comp hY)) |
| have hbound (n : ℕ) : ∀ᵐ t ∂τ, ∀ᵐ p ∂μ.prod (Measure.infinitePi ν), |
| ‖Z n (t, p)‖ ≤ 2 * ((d : ℝ) * beta) := by |
| filter_upwards with t |
| simpa only [Real.norm_eq_abs] using |
| hμ.finiteGibbsField_brownian_discrepancy_bound hbeta ν hB hmB t v n |
| have hzero : ∀ᵐ t ∂τ, ∀ᵐ p ∂μ.prod (Measure.infinitePi ν), |
| Tendsto (fun n => Z n (t, p)) atTop (𝓝 (0 : ℝ)) := by |
| filter_upwards with t |
| filter_upwards [hμ.finiteGibbsField_brownian_tendsto_ae hbeta ν hB hmB t v] |
| with p hp |
| simpa only [Z, Y, sub_self] using hp.sub_const |
| (Real.artanh (infiniteGibbsSpinMean beta v (Y (t, p)))) |
| have hlim := tendsto_integral_prod_sq_sub_of_bounded_ae τ |
| (μ.prod (Measure.infinitePi ν)) hZ (measurable_const (a := (0 : ℝ))) |
| (show 0 ≤ 2 * ((d : ℝ) * beta) by positivity) hbound |
| (by filter_upwards with t; filter_upwards with p; simp; positivity) hzero |
| simpa only [sub_zero] using hlim |
|
|
| end Problem367 |
|
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| end |
|
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| end OAI |
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|
