| import OAI.Probability.FreeIsing.Ising.CopyCavityBounds |
| import OAI.Probability.FreeIsing.Ising.CopySpectral |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
|
|
| namespace Problem367.IsingCopySpectral |
| open IsingObservationInnovation IsingObservationFactor IsingCopyStationarity PlantedSymmetry |
|
|
| variable {d : ℕ} {beta : ℝ} |
| variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ] |
|
|
| theorem measurable_field_at (beta : ℝ) (t : ℝ≥0) (v : TreeVertex d) : |
| Measurable (field beta v t) := by |
| have hm : Measurable (posterior beta v t) := |
| (measurable_infiniteGibbsSpinMean beta v).comp |
| (Measurable.of_eval (fun vertex => measurable_evaluation vertex t)) |
| unfold field Real.artanh |
| exact ((measurable_const.add hm).div (measurable_const.sub hm)).sqrt.log |
|
|
| theorem memLp_field_at |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (t : ℝ≥0) (v : TreeVertex d) : MemLp (field beta v t) 2 (observationLaw μ) := by |
| have hobs := (IsingConditionalCopyBounds.hasLaw_observationLaw_eval μ t).identDistrib |
| (show HasLaw (observation t) |
| ((μ.prod (gaussianNoiseLaw (TreeVertex d) t)).map (observation t)) |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) from |
| ⟨(measurable_observation t).aemeasurable, rfl⟩) |
| have ha : Measurable Real.artanh := by unfold Real.artanh; fun_prop |
| have hfield := hobs.comp (ha.comp (measurable_infiniteGibbsSpinMean beta v)) |
| apply hfield.memLp_iff.mpr |
| apply MemLp.ae_eq _ (hμ.memLp_gaussianPosteriorField_two hbeta t v) |
| filter_upwards [hμ.infiniteGibbsSpinMean_eq_posterior t v] with p hp |
| exact (congrArg Real.artanh hp).symm |
|
|
| def cavityObservable (hbeta : 0 ≤ beta) (t : ℝ≥0) (v w : TreeVertex d) |
| (x : Paths d) : ℝ := |
| CavityArray.field (tanh_nonneg_of_beta_nonneg hbeta) (Real.tanh_lt_one beta) |
| (fun i => field beta i t x) v w |
|
|
| theorem measurable_cavityObservable (hbeta : 0 ≤ beta) (t : ℝ≥0) (v w : TreeVertex d) : |
| Measurable (cavityObservable hbeta t v w) := |
| (CavityArray.measurable_field_apply (tanh_nonneg_of_beta_nonneg hbeta) |
| (Real.tanh_lt_one beta) v w).comp |
| (Measurable.of_eval (fun vertex => measurable_field_at beta t vertex)) |
|
|
| theorem memLp_cavityObservable |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (t : ℝ≥0) (v w : TreeVertex d) : |
| MemLp (cavityObservable hbeta t v w) 2 (observationLaw μ) := by |
| have hd : MemLp (fun x => cavityObservable hbeta t v w x - field beta v t x) 2 |
| (observationLaw μ) := |
| MemLp.of_bound |
| ((measurable_cavityObservable hbeta t v w).sub (measurable_field_at beta t v)).aestronglyMeasurable |
| (Real.artanh (Real.tanh beta)) (Filter.Eventually.of_forall (fun x => by |
| exact CavityArray.field_site_bound (tanh_nonneg_of_beta_nonneg hbeta) |
| (Real.tanh_lt_one beta) (fun i => field beta i t x) v w)) |
| have hs := hd.add (memLp_field_at hμ hbeta t v) |
| change MemLp (fun x => (cavityObservable hbeta t v w x - field beta v t x) + |
| field beta v t x) 2 (observationLaw μ) at hs |
| simpa only [sub_add_cancel] using hs |
|
|
| omit [IsProbabilityMeasure μ] in |
| theorem cavityObservable_relabel_ae |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (g : TreeAut d) (t : ℝ≥0) (v w : TreeVertex d) : |
| (fun x => cavityObservable hbeta t v w (relabel g x)) =ᵐ[observationLaw μ] |
| cavityObservable hbeta t (g.toEquiv.symm v) (g.toEquiv.symm w) := by |
| filter_upwards [field_array_relabel_ae hμ g t] with x hx |
| unfold cavityObservable |
| rw [hx] |
| rfl |
|
|
| def cavityLogObservable (hbeta : 0 ≤ beta) (t : ℝ≥0) (v w : TreeVertex d) |
| (x : Paths d) : ℝ := |
| Real.log (responseAttenuation (Real.tanh beta) (cavityObservable hbeta t v w x)) |
|
|
| theorem memLp_cavityLogObservable |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (t : ℝ≥0) (v w : TreeVertex d) : |
| MemLp (cavityLogObservable hbeta t v w) 2 (observationLaw μ) := by |
| apply memLp_log_responseAttenuation _ (memLp_cavityObservable hμ hbeta t v w) |
| have h0 := tanh_nonneg_of_beta_nonneg hbeta |
| have h1 := Real.tanh_lt_one beta |
| nlinarith |
|
|
| def observableDifference {f : Paths d → ℝ} (hf : MemLp f 2 (observationLaw μ)) : |
| differenceSpace beta μ := |
| CopyDifference.toRangeClosure (CopyDifference.operator |
| (measurePreserving_observations (beta := beta) (μ := μ)).1 |
| (measurePreserving_observations (beta := beta) (μ := μ)).2) (hf.toLp f) |
|
|
| theorem observableDifference_ae {f : Paths d → ℝ} (hf : MemLp f 2 (observationLaw μ)) : |
| (observableDifference (beta := beta) hf : Lp ℝ 2 (copies beta μ)) =ᵐ[copies beta μ] |
| (fun q => f q.2.1 - f q.2.2) := by |
| obtain ⟨hp, hq⟩ := measurePreserving_observations (beta := beta) (μ := μ) |
| have h := CopyDifference.operator_ae hp hq (hf.toLp f) |
| filter_upwards [h, hp.quasiMeasurePreserving.ae hf.coeFn_toLp, |
| hq.quasiMeasurePreserving.ae hf.coeFn_toLp] with q hq' hl hr |
| exact hq'.trans (congrArg₂ (fun a b : ℝ => a - b) hl hr) |
|
|
| theorem norm_observableDifference_sq {f : Paths d → ℝ} (hf : MemLp f 2 (observationLaw μ)) : |
| ‖observableDifference (beta := beta) hf‖ ^ 2 = |
| ∫ q, (f q.2.1 - f q.2.2) ^ 2 ∂copies beta μ := by |
| change ‖(observableDifference (beta := beta) hf : Lp ℝ 2 (copies beta μ))‖ ^ 2 = _ |
| rw [← real_inner_self_eq_norm_sq, L2.inner_def] |
| apply integral_congr_ae |
| filter_upwards [observableDifference_ae (beta := beta) hf] with q hq |
| simp [hq, pow_two] |
|
|
| theorem norm_cavityLogDifference_sq_le |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (t : ℝ≥0) (o v w : TreeVertex d) : |
| ‖observableDifference (beta := beta) (memLp_cavityLogObservable hμ hbeta t v w)‖ ^ 2 ≤ |
| cavityLogVarianceCoefficient (Real.tanh beta) * |
| ∫ q, (IsingConditionalCopyBounds.fieldDifference beta o t q) ^ 2 ∂copies beta μ := by |
| rw [norm_observableDifference_sq] |
| have h := integral_sitePair_log_cavity_sub_sq_le hμ hbeta t o v w |
| have hm (a b : TreeVertex d) : Measurable (fun p : CavityArray.SitePair d => |
| (Real.log (responseAttenuation (Real.tanh beta) |
| (CavityArray.field (tanh_nonneg_of_beta_nonneg hbeta) (Real.tanh_lt_one beta) p.1 a b)) - |
| Real.log (responseAttenuation (Real.tanh beta) |
| (CavityArray.field (tanh_nonneg_of_beta_nonneg hbeta) (Real.tanh_lt_one beta) p.2 a b))) ^ 2) := by |
| have ht : (Real.tanh beta) ^ 2 ≤ 1 := by |
| have h0 := tanh_nonneg_of_beta_nonneg hbeta |
| have h1 := Real.tanh_lt_one beta |
| nlinarith |
| have hl := (lipschitzWith_log_responseAttenuation ht).continuous.measurable |
| have hc := CavityArray.measurable_field_apply (tanh_nonneg_of_beta_nonneg hbeta) |
| (Real.tanh_lt_one beta) a b |
| exact ((hl.comp (hc.comp measurable_fst)).sub (hl.comp (hc.comp measurable_snd))).pow_const 2 |
| have hs : Measurable (fun p : CavityArray.SitePair d => (p.1 o - p.2 o) ^ 2) := |
| (((measurable_pi_apply o).comp measurable_fst).sub |
| ((measurable_pi_apply o).comp measurable_snd)).pow_const 2 |
| dsimp only at h |
| rw [integral_map (measurable_sitePair beta t).aemeasurable (hm v w).aestronglyMeasurable, |
| integral_map (measurable_sitePair beta t).aemeasurable hs.aestronglyMeasurable] at h |
| exact h |
|
|
| end Problem367.IsingCopySpectral |
|
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| end |
|
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| end OAI |
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