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f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 | 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
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