| import OAI.Probability.FreeIsing.Cavity.Array |
| import OAI.Probability.FreeIsing.Estimates.ThresholdArithmetic |
| import OAI.Probability.FreeIsing.Posterior.FieldL2Limit |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped Topology NNReal |
|
|
| namespace Problem367 |
|
|
| open PlantedSymmetry |
|
|
| namespace CavityFieldLimit |
|
|
| local instance : Nonempty IsingSpin := ⟨.neg⟩ |
|
|
| def finiteSiteField {d : ℕ} (beta : ℝ) (D : Finset (TreeVertex d)) |
| (v : D) (y : TreeVertex d → ℝ) : ℝ := |
| Real.artanh (FinitePosterior.fieldMean (finiteIsingEnergy beta D) |
| (fun (eta : D → IsingSpin) w => spinSign (eta w)) |
| (fun eta => spinSign (eta v)) (D.restrict y)) |
|
|
| private theorem measurable_artanh : Measurable Real.artanh := by |
| unfold Real.artanh |
| fun_prop |
|
|
| theorem measurable_finiteSiteField {d : ℕ} (beta : ℝ) |
| (D : Finset (TreeVertex d)) (v : D) : Measurable (finiteSiteField beta D v) := |
| measurable_artanh.comp ((FinitePosterior.continuous_fieldMean _ _ _).measurable.comp |
| D.measurable_restrict) |
|
|
| def finiteCavityField {d : ℕ} {beta : ℝ} (hbeta : 0 ≤ beta) |
| (D : Finset (TreeVertex d)) (v w : D) (y : TreeVertex d → ℝ) : ℝ := |
| (cavityFromSites (tanh_nonneg_of_beta_nonneg hbeta) (Real.tanh_lt_one beta) |
| (finiteSiteField beta D v y, finiteSiteField beta D w y)).1 |
|
|
| def infiniteCavityField {d : ℕ} {beta : ℝ} (hbeta : 0 ≤ beta) |
| (v w : TreeVertex d) (y : TreeVertex d → ℝ) : ℝ := |
| CavityArray.field (tanh_nonneg_of_beta_nonneg hbeta) (Real.tanh_lt_one beta) |
| (fun u => Real.artanh (infiniteGibbsSpinMean beta u y)) v w |
|
|
| theorem measurable_finiteCavityField {d : ℕ} {beta : ℝ} (hbeta : 0 ≤ beta) |
| (D : Finset (TreeVertex d)) (v w : D) : |
| Measurable (finiteCavityField hbeta D v w) := |
| measurable_fst.comp ((measurable_cavityFromSites _ _).comp |
| ((measurable_finiteSiteField beta D v).prodMk (measurable_finiteSiteField beta D w))) |
|
|
| theorem measurable_infiniteCavityField {d : ℕ} {beta : ℝ} (hbeta : 0 ≤ beta) |
| (v w : TreeVertex d) : Measurable (infiniteCavityField hbeta v w) := by |
| exact (CavityArray.measurable_field_apply _ _ v w).comp |
| (Measurable.of_eval (fun u => measurable_artanh.comp |
| (measurable_infiniteGibbsSpinMean beta u))) |
|
|
| end CavityFieldLimit |
|
|
| open CavityFieldLimit |
|
|
| theorem HasFreeIsingLaw.exhaustionSiteField_tendsto_ae |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (D : ℕ → Finset (TreeVertex d)) (hD : Monotone D) |
| (hconn : ∀ n, ConnectedFinset (D n)) |
| (hex : ∀ u, ∃ n, u ∈ D n) (v : TreeVertex d) (hv : ∀ n, v ∈ D n) |
| (t : ℝ≥0) : |
| ∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t), |
| Tendsto (fun n => finiteSiteField beta (D n) ⟨v, hv 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 (D n) (hconn n) t ⟨v, hv n⟩ |
| have hlim := tendsto_posterior_fields_ae |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) (finiteObservationFiltration D hD t) |
| (fun p => spinSign (p.1 v)) (fun p => observation t p v) ((d : ℝ) * beta) hb |
| rw [finiteObservationFiltration_iSup D hD hex t] at hlim |
| have hce (n : ℕ) := hμ.finiteObservation_condExp (D n) (hconn n) t |
| (fun eta => spinSign (eta ⟨v, hv 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.exhaustionSiteField_tendsto_meanSquare |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (D : ℕ → Finset (TreeVertex d)) (hD : Monotone D) |
| (hconn : ∀ n, ConnectedFinset (D n)) |
| (hex : ∀ u, ∃ n, u ∈ D n) (v : TreeVertex d) (hv : ∀ n, v ∈ D n) |
| (t : ℝ≥0) : |
| Tendsto (fun n => ∫ p, |
| (finiteSiteField beta (D n) ⟨v, hv 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 (D n) (hconn n) t ⟨v, hv n⟩ |
| have hlim := tendsto_integral_sq_sub_posterior_fields |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) (finiteObservationFiltration D hD t) |
| (fun p => spinSign (p.1 v)) (fun p => observation t p v) hb |
| rw [finiteObservationFiltration_iSup D hD hex t] at hlim |
| convert hlim using 1 |
| ext n |
| apply integral_congr_ae |
| filter_upwards [hμ.finiteObservation_condExp (D n) (hconn n) t |
| (fun eta => spinSign (eta ⟨v, hv 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.exhaustionCavityField_tendsto_ae |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (D : ℕ → Finset (TreeVertex d)) (hD : Monotone D) |
| (hconn : ∀ n, ConnectedFinset (D n)) (hex : ∀ u, ∃ n, u ∈ D n) |
| (v w : TreeVertex d) (hv : ∀ n, v ∈ D n) (hw : ∀ n, w ∈ D n) |
| (t : ℝ≥0) : |
| ∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t), |
| Tendsto (fun n => finiteCavityField hbeta (D n) ⟨v, hv n⟩ ⟨w, hw n⟩ |
| (observation t p)) atTop (𝓝 (infiniteCavityField hbeta v w (observation t p))) := by |
| filter_upwards [hμ.exhaustionSiteField_tendsto_ae hbeta D hD hconn hex v hv t, |
| hμ.exhaustionSiteField_tendsto_ae hbeta D hD hconn hex w hw t] with p hp hq |
| exact (continuous_fst.comp (continuous_cavityFromSites |
| (tanh_nonneg_of_beta_nonneg hbeta) (Real.tanh_lt_one beta))).continuousAt.tendsto.comp |
| (hp.prodMk_nhds hq) |
|
|
| theorem HasFreeIsingLaw.exhaustionCavityField_discrepancy_bound |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (D : Finset (TreeVertex d)) (v w : D) (t : ℝ≥0) : |
| ∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t), |
| |finiteCavityField hbeta D v w (observation t p) - |
| infiniteCavityField hbeta v.1 w.1 (observation t p)| ≤ |
| (2 * ((d : ℝ) * beta)) / (1 - Real.tanh beta) := by |
| filter_upwards [hμ.gaussian_posterior_field_bound hbeta t v.1, |
| hμ.gaussian_posterior_field_bound hbeta t w.1, |
| hμ.infiniteGibbsSpinMean_eq_posterior t v.1, |
| hμ.infiniteGibbsSpinMean_eq_posterior t w.1] with p hbv hbw hev hew |
| let y := observation t p |
| have hb (u : D) (hu : |Real.artanh (infiniteGibbsSpinMean beta u.1 y) - y u.1| ≤ |
| (d : ℝ) * beta) : |
| |finiteSiteField beta D u y - Real.artanh (infiniteGibbsSpinMean beta u.1 y)| ≤ |
| 2 * ((d : ℝ) * beta) := by |
| have hf := finite_tree_gibbs_field_uniform_bound D hbeta (D.restrict y) u |
| change |finiteSiteField beta D u y - y u.1| ≤ (d : ℝ) * beta at hf |
| calc |
| _ = |(finiteSiteField beta D u y - y u.1) - |
| (Real.artanh (infiniteGibbsSpinMean beta u.1 y) - y u.1)| := by congr 1; ring |
| _ ≤ |finiteSiteField beta D u y - y u.1| + |
| |Real.artanh (infiniteGibbsSpinMean beta u.1 y) - y u.1| := abs_sub _ _ |
| _ ≤ _ := by linarith |
| have hv := hb v (by simpa only [y, hev] using hbv.2) |
| have hw := hb w (by simpa only [y, hew] using hbw.2) |
| have h0 := tanh_nonneg_of_beta_nonneg hbeta |
| have h1 := Real.tanh_lt_one beta |
| have hs := cavityFromSites_spec h0 h1 (finiteSiteField beta D v y, finiteSiteField beta D w y) |
| have hs' := cavityFromSites_spec h0 h1 |
| (Real.artanh (infiniteGibbsSpinMean beta v.1 y), Real.artanh (infiniteGibbsSpinMean beta w.1 y)) |
| exact cavity_pair_sub_abs_le h0 h1 hs.1 hs.2 hs'.1 hs'.2 hv hw |
|
|
| theorem HasFreeIsingLaw.exhaustionCavityField_tendsto_meanSquare |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (D : ℕ → Finset (TreeVertex d)) (hD : Monotone D) |
| (hconn : ∀ n, ConnectedFinset (D n)) (hex : ∀ u, ∃ n, u ∈ D n) |
| (v w : TreeVertex d) (hv : ∀ n, v ∈ D n) (hw : ∀ n, w ∈ D n) |
| (t : ℝ≥0) : |
| Tendsto (fun n => ∫ p, |
| (finiteCavityField hbeta (D n) ⟨v, hv n⟩ ⟨w, hw n⟩ (observation t p) - |
| infiniteCavityField hbeta v w (observation t p)) ^ 2 |
| ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) atTop (𝓝 0) := by |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| have hm (n : ℕ) := ((measurable_finiteCavityField hbeta (D n) ⟨v, hv n⟩ ⟨w, hw n⟩).comp |
| (measurable_observation t)).sub ((measurable_infiniteCavityField hbeta v w).comp |
| (measurable_observation t)) |
| have hbound (n : ℕ) : ∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t), |
| ‖(finiteCavityField hbeta (D n) ⟨v, hv n⟩ ⟨w, hw n⟩ (observation t p) - |
| infiniteCavityField hbeta v w (observation t p)) ^ 2‖ ≤ |
| ((2 * ((d : ℝ) * beta)) / (1 - Real.tanh beta)) ^ 2 := by |
| filter_upwards [hμ.exhaustionCavityField_discrepancy_bound hbeta (D n) |
| ⟨v, hv n⟩ ⟨w, hw n⟩ t] with p hp |
| rw [Real.norm_of_nonneg (sq_nonneg _)] |
| simpa only [sq_abs] using pow_le_pow_left₀ (abs_nonneg _) hp 2 |
| have hzero := hμ.exhaustionCavityField_tendsto_ae hbeta D hD hconn hex v w hv hw t |
| have hlim := tendsto_integral_of_dominated_convergence |
| (fun _ => ((2 * ((d : ℝ) * beta)) / (1 - Real.tanh beta)) ^ 2) |
| (fun n => (hm n).aestronglyMeasurable.pow 2) (integrable_const _) hbound |
| (hzero.mono (fun p hp => by simpa using |
| (hp.sub_const (infiniteCavityField hbeta v w (observation t p))).pow 2)) |
| simpa using hlim |
|
|
| end Problem367 |
|
|
| end |
|
|
| end OAI |
|
|
