| import OAI.Probability.FreeIsing.Critical.CanonicalRowEnergy |
| import OAI.Probability.FreeIsing.Posterior.ArrayPosteriorCovarianceStationarity |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal |
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| namespace Problem367 |
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| open PlantedSymmetry CriticalCanonicalRowEnergy |
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| theorem HasFreeIsingLaw.gaussian_posteriorResponse_relabel_ae |
| {d : ℕ} {theta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw theta μ) (t : ℝ≥0) (g : TreeAut d) |
| (u v : TreeVertex d) : |
| gaussianPosteriorResponse μ t u v ∘ Prod.map (relabel g) (relabel g) =ᵐ[ |
| μ.prod (gaussianNoiseLaw (TreeVertex d) t)] |
| gaussianPosteriorResponse μ t (g.toEquiv.symm u) (g.toEquiv.symm v) := by |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| let F : IsingSpin → ℝ → ℝ := fun s z => (t : ℝ) * spinSign s + z |
| have hfield : (observationSigma (TreeVertex d) (t : ℝ)).comap |
| (Prod.map (relabel g) (relabel g)) = observationSigma (TreeVertex d) (t : ℝ) := by |
| change (selectedObservationField F id).comap |
| (Prod.map (relabel g) (relabel g)) = selectedObservationField F id |
| rw [selectedObservationField_comap_relabel] |
| apply selectedObservationField_eq_of_range_eq |
| simp only [id_eq, Equiv.range_eq_univ, Set.range_id] |
| have hs : Measurable spinSign := by intro s hs; trivial |
| have hm (w : TreeVertex d) : Measurable |
| (fun p : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) => spinSign (p.1 w)) := |
| hs.comp ((measurable_pi_apply w).comp measurable_fst) |
| have h := posteriorResponse_comp_measurePreserving |
| (PlantedStationarity.measurePreserving_planted_relabel hμ (gaussianReal 0 t) g) |
| (hm u) (hm v) (integrable_plantedSpin _ u) (integrable_plantedSpin _ v) |
| (ArrayPosteriorStationarity.integrable_spin_pair _ u v) |
| (observationSigma_le (TreeVertex d) (t : ℝ)) |
| rw [hfield] at h |
| exact h.symm |
|
|
| theorem HasFreeIsingLaw.gaussian_response_critical_intrinsic_bound |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d) |
| (hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1) |
| {t : ℝ≥0} (ht : 0 < t) (u v : TreeVertex d) : |
| (∫ p, gaussianPosteriorResponse μ t u v p ^ 2 |
| ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤ |
| (Real.tanh beta ^ 2 * Real.exp (-rate beta t)) ^ reducedWordDistance u.1 v.1 := by |
| obtain ⟨g, hgu, hdist⟩ := exists_treeAut_to_root_distance u |
| change g.toEquiv u = ConnectedHull.root d at hgu |
| have hsu : g.toEquiv.symm (ConnectedHull.root d) = u := by |
| rw [← hgu, Equiv.symm_apply_apply] |
| have hn := hμ.gaussian_posteriorResponse_relabel_ae t g (ConnectedHull.root d) (g.toEquiv v) |
| simp only [hsu, Equiv.symm_apply_apply] at hn |
| have hmp : MeasurePreserving (Prod.map (relabel g) (relabel g)) |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) := |
| PlantedStationarity.measurePreserving_planted_relabel hμ (gaussianReal 0 t) g |
| have hi := (hμ.gaussian_posteriorResponse_sq_decay hb ht |
| (ConnectedHull.root d) (g.toEquiv v) rfl).1 |
| calc |
| _ = ∫ p, gaussianPosteriorResponse μ t (ConnectedHull.root d) (g.toEquiv v) |
| (Prod.map (relabel g) (relabel g) p) ^ 2 |
| ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t) := |
| integral_congr_ae (hn.symm.fun_comp (fun x : ℝ => x ^ 2)) |
| _ = ∫ p, gaussianPosteriorResponse μ t (ConnectedHull.root d) (g.toEquiv v) p ^ 2 |
| ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t) := |
| hmp.hasLaw.integral_comp hi.aestronglyMeasurable |
| _ ≤ _ := by |
| simpa only [hdist] using |
| hμ.gaussian_rootResponse_critical_radial_bound hb hd hc ht (g.toEquiv v) |
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| theorem HasFreeIsingLaw.gaussian_response_critical_row_energy |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d) |
| (hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1) |
| {t : ℝ≥0} (ht : 0 < t) (u : TreeVertex d) : |
| (∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t), |
| Summable (fun v => gaussianPosteriorResponse μ t u v p ^ 2)) ∧ |
| Integrable (fun p => ∑' v, gaussianPosteriorResponse μ t u v p ^ 2) |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ∧ |
| (∫ p, ∑' v, gaussianPosteriorResponse μ t u v p ^ 2 |
| ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤ |
| 1 + (d : ℝ) * Real.tanh beta ^ 2 / rate beta t := by |
| obtain ⟨g, hgu⟩ := exists_treeAut_to_root u |
| change g.toEquiv u = ConnectedHull.root d at hgu |
| have hsu : g.toEquiv.symm (ConnectedHull.root d) = u := by |
| rw [← hgu, Equiv.symm_apply_apply] |
| have hn := ae_all_iff.mpr (fun v => |
| hμ.gaussian_posteriorResponse_relabel_ae t g (ConnectedHull.root d) v) |
| simp only [hsu] at hn |
| have hmp : MeasurePreserving (Prod.map (relabel g) (relabel g)) |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) := |
| PlantedStationarity.measurePreserving_planted_relabel hμ (gaussianReal 0 t) g |
| have hg := hμ.gaussian_rootResponse_critical_row_energy hb hd hc ht |
| have he : (fun p => ∑' v, |
| gaussianPosteriorResponse μ t (ConnectedHull.root d) v p ^ 2) ∘ |
| Prod.map (relabel g) (relabel g) =ᵐ[ |
| μ.prod (gaussianNoiseLaw (TreeVertex d) t)] |
| fun p => ∑' v, gaussianPosteriorResponse μ t u v p ^ 2 := by |
| filter_upwards [hn] with p hp |
| change (∑' v, gaussianPosteriorResponse μ t (ConnectedHull.root d) v |
| (Prod.map (relabel g) (relabel g) p) ^ 2) = _ |
| exact (tsum_congr (fun v => congrArg (fun x : ℝ => x ^ 2) (hp v))).trans |
| (g.toEquiv.symm.tsum_eq (fun v => gaussianPosteriorResponse μ t u v p ^ 2)) |
| refine ⟨?_, (hmp.integrable_comp_of_integrable hg.2.1).congr he, ?_⟩ |
| · filter_upwards [hmp.quasiMeasurePreserving.ae hg.1, hn] with p hp hnat |
| exact g.toEquiv.symm.summable_iff.mp |
| (hp.congr (fun v => congrArg (fun x : ℝ => x ^ 2) (hnat v))) |
| · rw [← integral_congr_ae he, hmp.hasLaw.integral_comp hg.2.1.aestronglyMeasurable] |
| exact hμ.integral_gaussian_rootResponse_row_le_inv_rate hb hd hc ht |
|
|
| theorem HasFreeIsingLaw.infiniteGibbsResponse_eq_gaussian_response |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (t : ℝ≥0) (u v : TreeVertex d) : |
| (fun p => infiniteGibbsResponse beta u v (observation t p)) =ᵐ[ |
| μ.prod (gaussianNoiseLaw (TreeVertex d) t)] gaussianPosteriorResponse μ t u v := by |
| filter_upwards [hμ.infiniteGibbsCovariance_eq_posterior t u v, |
| hμ.infiniteGibbsSpinMean_eq_posterior t u] with p hc hm |
| unfold infiniteGibbsResponse gaussianPosteriorResponse posteriorResponse |
| rw [hc, hm] |
| rfl |
|
|
| theorem HasFreeIsingLaw.infiniteGibbsResponse_critical_row_energy |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d) |
| (hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1) |
| {t : ℝ≥0} (ht : 0 < t) (u : TreeVertex d) : |
| (∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t), |
| Summable (fun v => infiniteGibbsResponse beta u v (observation t p) ^ 2)) ∧ |
| Integrable (fun p => infiniteGibbsResponseRow beta u (observation t p)) |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ∧ |
| (∫ p, infiniteGibbsResponseRow beta u (observation t p) |
| ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤ |
| 1 + (d : ℝ) * Real.tanh beta ^ 2 / rate beta t := by |
| have hg := hμ.gaussian_response_critical_row_energy hb hd hc ht u |
| have hn := ae_all_iff.mpr (fun v => hμ.infiniteGibbsResponse_eq_gaussian_response t u v) |
| have he : (fun p => infiniteGibbsResponseRow beta u (observation t p)) =ᵐ[ |
| μ.prod (gaussianNoiseLaw (TreeVertex d) t)] |
| fun p => ∑' v, gaussianPosteriorResponse μ t u v p ^ 2 := by |
| filter_upwards [hn] with p hp |
| exact tsum_congr fun v => congrArg (fun x : ℝ => x ^ 2) (hp v) |
| refine ⟨?_, hg.2.1.congr he.symm, ?_⟩ |
| · filter_upwards [hg.1, hn] with p hp hnat |
| exact hp.congr fun v => congrArg (fun x : ℝ => x ^ 2) (hnat v).symm |
| · rw [integral_congr_ae he] |
| exact hg.2.2 |
|
|
| theorem HasFreeIsingLaw.brownian_response_critical_row_energy |
| {d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω] |
| {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d) |
| (hc : ((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)) {t : ℝ≥0} (ht : 0 < t) (u : TreeVertex d) : |
| (∀ᵐ p ∂μ.prod (Measure.infinitePi ν), |
| Summable (fun v => brownianPosteriorResponse μ ν B t u v p ^ 2)) ∧ |
| Integrable (fun p => ∑' v, brownianPosteriorResponse μ ν B t u v p ^ 2) |
| (μ.prod (Measure.infinitePi ν)) ∧ |
| (∫ p, ∑' v, brownianPosteriorResponse μ ν B t u v p ^ 2 |
| ∂μ.prod (Measure.infinitePi ν)) ≤ |
| 1 + (d : ℝ) * Real.tanh beta ^ 2 / rate beta t := by |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| have (i : TreeVertex d) : IsProbabilityMeasure (ν i) := |
| (hB i).isGaussianProcess.isProbabilityMeasure |
| have hT := (MeasurePreserving.id μ).prod |
| (brownian_array_eval_measurePreserving ν hB hmB t) |
| have hg := hμ.gaussian_response_critical_row_energy hb hd hc ht u |
| have heq := ae_all_iff.mpr (fun v => |
| brownian_past_posteriorResponse_eq_gaussian μ ν hB hmB t u v) |
| have he : (fun p => ∑' v, brownianPosteriorResponse μ ν B t u v p ^ 2) =ᵐ[ |
| μ.prod (Measure.infinitePi ν)] |
| (fun q => ∑' v, gaussianPosteriorResponse μ t u v q ^ 2) ∘ |
| Prod.map id (fun p i => B i t (p i)) := by |
| filter_upwards [heq] with p hp |
| exact tsum_congr fun v => congrArg (fun x : ℝ => x ^ 2) (hp v) |
| refine ⟨?_, (hT.integrable_comp_of_integrable hg.2.1).congr he.symm, ?_⟩ |
| · filter_upwards [hT.quasiMeasurePreserving.ae hg.1, heq] with p hp hpeq |
| exact hp.congr fun v => congrArg (fun x : ℝ => x ^ 2) (hpeq v).symm |
| · rw [integral_congr_ae he, hT.hasLaw.integral_comp hg.2.1.aestronglyMeasurable] |
| exact hg.2.2 |
|
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| end Problem367 |
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| end |
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| end OAI |
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