| import OAI.Probability.FreeIsing.Cavity.FieldL2Limit | |
| import OAI.Probability.FreeIsing.Estimates.FiniteComponentFields | |
| namespace OAI | |
| noncomputable section | |
| namespace Problem367.CavityFieldLimit | |
| variable {d : ℕ} | |
| lemma finiteSiteField_eq_component_field (beta : ℝ) (D : Finset (TreeVertex d)) | |
| (v : D) (y : TreeVertex d → ℝ) : | |
| finiteSiteField beta D v y = FiniteIsingSplit.field beta D v y := by | |
| simp only [finiteSiteField, FiniteIsingSplit.field, FiniteIsingSplit.mean_eq_fieldMean] | |
| theorem finiteCavityField_root_eq_complement {beta : ℝ} (hbeta : 0 ≤ beta) | |
| (D : Finset (TreeVertex d)) (hroot : ConnectedHull.root d ∈ D) | |
| (c : Fin d) (hc : (⟨[c], List.IsChain.singleton c⟩ : TreeVertex d) ∈ D) | |
| (y : TreeVertex d → ℝ) : | |
| finiteCavityField hbeta D ⟨ConnectedHull.root d, hroot⟩ | |
| ⟨⟨[c], List.IsChain.singleton c⟩, hc⟩ y = | |
| finiteSiteField beta (rootComplementFinset D c) | |
| ⟨ConnectedHull.root d, rootComplementFinset_root_mem c hroot⟩ y := by | |
| simp only [finiteCavityField, finiteSiteField_eq_component_field] | |
| exact congrArg Prod.fst | |
| (FiniteIsingSplit.cavityFromSites_root_component hbeta D hroot c hc y) | |
| theorem finiteCavityField_child_eq_component {beta : ℝ} (hbeta : 0 ≤ beta) | |
| (D : Finset (TreeVertex d)) (hroot : ConnectedHull.root d ∈ D) | |
| (c : Fin d) (hc : (⟨[c], List.IsChain.singleton c⟩ : TreeVertex d) ∈ D) | |
| (y : TreeVertex d → ℝ) : | |
| finiteCavityField hbeta D ⟨⟨[c], List.IsChain.singleton c⟩, hc⟩ | |
| ⟨ConnectedHull.root d, hroot⟩ y = | |
| finiteSiteField beta (rootComponentFinset D c) | |
| ⟨⟨[c], List.IsChain.singleton c⟩, rootComponentFinset_root_mem c hc⟩ y := by | |
| simp only [finiteCavityField, finiteSiteField_eq_component_field] | |
| have hswap := cavityFromSites_swap (tanh_nonneg_of_beta_nonneg hbeta) | |
| (Real.tanh_lt_one beta) | |
| (FiniteIsingSplit.field beta D ⟨ConnectedHull.root d, hroot⟩ y, | |
| FiniteIsingSplit.field beta D ⟨⟨[c], List.IsChain.singleton c⟩, hc⟩ y) | |
| rw [FiniteIsingSplit.cavityFromSites_root_component hbeta D hroot c hc y] at hswap | |
| exact congrArg Prod.fst hswap | |
| end Problem367.CavityFieldLimit | |
| end | |
| end OAI | |