| import OAI.Probability.FreeIsing.Ising.CopyCavityBounds |
| import OAI.Probability.FreeIsing.Trees.OrbitAverage |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory |
| open scoped NNReal |
|
|
| namespace Problem367.IsingCopyOrbitAverage |
|
|
| open IsingObservationInnovation IsingObservationFactor IsingConditionalCopyBounds |
| open TreeOrbitAverage |
|
|
| variable {d : ℕ} {beta : ℝ} |
| variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ] |
|
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| def copyAction (g : TreeAut d) : CopySpace d ≃ᵐ CopySpace d := |
| let e := ObservationCopyStationarity.arrayRelabelEquiv (A := WienerSampler.Path) g.symm |
| e.prodCongr (e.prodCongr e) |
|
|
| theorem measurePreserving_copyAction |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (g : TreeAut d) : |
| MeasurePreserving (copyAction g) (copies beta μ) (copies beta μ) := |
| IsingCopyStationarity.measurePreserving_copies_relabel hμ g.symm |
|
|
| theorem fieldDifference_copyAction_ae |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (g : TreeAut d) |
| (t : ℝ≥0) (v : TreeVertex d) : |
| (fun q => fieldDifference beta v t (copyAction g q)) =ᵐ[copies beta μ] |
| fieldDifference beta (g.toEquiv v) t := by |
| obtain ⟨hl, hr⟩ := IsingCopyStationarity.measurePreserving_observations |
| (beta := beta) (μ := μ) |
| filter_upwards [hl.quasiMeasurePreserving.ae |
| (IsingCopyStationarity.field_array_relabel_ae hμ g.symm t), |
| hr.quasiMeasurePreserving.ae |
| (IsingCopyStationarity.field_array_relabel_ae hμ g.symm t)] with q hq hq' |
| change field beta v t (relabel g.symm q.2.1) - |
| field beta v t (relabel g.symm q.2.2) = |
| field beta (g.toEquiv v) t q.2.1 - field beta (g.toEquiv v) t q.2.2 |
| rw [congrFun hq v, congrFun hq' v] |
| rfl |
|
|
| theorem fieldDifference_sq_invariant |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (t : ℝ≥0) |
| {stem : List (Fin d)} (o : TreeVertex d) (ho : o.1.length ≤ stem.length) |
| (g : TreeAut d) (hg : FixesStemBall stem g) : |
| ∀ᵐ q ∂copies beta μ, |
| (fieldDifference beta o t (copyAction g q)) ^ 2 = |
| (fieldDifference beta o t q) ^ 2 := by |
| filter_upwards [fieldDifference_copyAction_ae hμ g t o] with q hq |
| rw [hq, hg o ho] |
|
|
| theorem integral_fieldDifference_sq_mul_extension_average |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (t : ℝ≥0) {stem : List (Fin d)} {n : ℕ} |
| (o : TreeVertex d) (ho : o.1.length ≤ stem.length) (s₀ : Extension stem n) |
| (F : TreeVertex d → CopySpace d → ℝ) |
| (hmF : ∀ v, Measurable (F v)) |
| (hF01 : ∀ v q, 0 ≤ F v q ∧ F v q ≤ 1) |
| (hcov : ∀ g, FixesStemBall stem g → ∀ v, ∀ᵐ q ∂copies beta μ, |
| F v (copyAction g q) = F (g.toEquiv v) q) : |
| (∫ q, (fieldDifference beta o t q) ^ 2 * F (endpoint s₀) q ∂copies beta μ) = |
| ∫ q, (fieldDifference beta o t q) ^ 2 * WeightedOrbitAverage.average |
| (fun s : Extension stem n => F (endpoint s)) q ∂copies beta μ := by |
| apply TreeOrbitAverage.integral_weight_mul_extension_average s₀ copyAction |
| (measurePreserving_copyAction hμ) (fun q => (fieldDifference beta o t q) ^ 2) F |
| (fieldDifference_sq_invariant hμ t o ho) hcov |
| apply (IsingCopyStationarity.integrable_fieldDifference_sq hμ hbeta t o).mul_bdd |
| (hmF (endpoint s₀)).aestronglyMeasurable |
| exact Filter.Eventually.of_forall (fun q => by |
| simpa only [Real.norm_eq_abs, abs_of_nonneg (hF01 (endpoint s₀) q).1] |
| using (hF01 (endpoint s₀) q).2) |
|
|
| end Problem367.IsingCopyOrbitAverage |
|
|
| end |
|
|
| end OAI |
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|
