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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 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 | import OAI.Probability.FreeIsing.Estimates.FiniteComponentGeometry
import OAI.Probability.FreeIsing.Estimates.FiniteIsingSplit
namespace OAI
noncomputable section
namespace Problem367.FiniteIsingSplit
variable {d : ℕ}
lemma field_congr_domain (beta : ℝ) {D E : Finset (TreeVertex d)}
(h : D = E) (v : D) (w : E) (hv : v.1 = w.1)
(y : TreeVertex d → ℝ) : field beta D v y = field beta E w y := by
subst E
have hw : v = w := Subtype.ext hv
subst w
rfl
theorem cavityFromSites_root_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) :
cavityFromSites (tanh_nonneg_of_beta_nonneg hbeta) (Real.tanh_lt_one beta)
(field beta D ⟨ConnectedHull.root d, hroot⟩ y,
field beta D ⟨⟨[c], List.IsChain.singleton c⟩, hc⟩ y) =
(field beta (rootComplementFinset D c)
⟨ConnectedHull.root d, rootComplementFinset_root_mem c hroot⟩ y,
field beta (rootComponentFinset D c)
⟨⟨[c], List.IsChain.singleton c⟩, rootComponentFinset_root_mem c hc⟩ y) := by
classical
let A := rootComplementFinset D c
let B := rootComponentFinset D c
let u : A := ⟨ConnectedHull.root d, rootComplementFinset_root_mem c hroot⟩
let v : B := ⟨⟨[c], List.IsChain.singleton c⟩, rootComponentFinset_root_mem c hc⟩
have hAB : Disjoint A B := (rootComponentFinset_disjoint_complement D c).symm
have hUnion : A ∪ B = D := by
rw [Finset.union_comm]
exact rootComponentFinset_union_complement D c
have hEdges : inducedEdges (A ∪ B) =
insert (u.1, v.1) (inducedEdges A ∪ inducedEdges B) := by
rw [hUnion, inducedEdges_rootComponent_split hroot c hc]
congr 1
exact Finset.union_comm _ _
have h := cavityFromSites_eq_component_fields hbeta A B hAB u v hEdges y
have hu := field_congr_domain beta hUnion
(⟨u, Finset.mem_union_left B u.property⟩ : ↥(A ∪ B))
⟨ConnectedHull.root d, hroot⟩ rfl y
have hv := field_congr_domain beta hUnion
(⟨v, Finset.mem_union_right A v.property⟩ : ↥(A ∪ B))
⟨⟨[c], List.IsChain.singleton c⟩, hc⟩ rfl y
rw [hu, hv] at h
exact h
theorem field_root_eq_components (beta : ℝ)
(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) :
field beta D ⟨ConnectedHull.root d, hroot⟩ y =
field beta (rootComplementFinset D c)
⟨ConnectedHull.root d, rootComplementFinset_root_mem c hroot⟩ y +
cavityMessage (Real.tanh beta) (field beta (rootComponentFinset D c)
⟨⟨[c], List.IsChain.singleton c⟩, rootComponentFinset_root_mem c hc⟩ y) := by
classical
let A := rootComplementFinset D c
let B := rootComponentFinset D c
let u : A := ⟨ConnectedHull.root d, rootComplementFinset_root_mem c hroot⟩
let v : B := ⟨⟨[c], List.IsChain.singleton c⟩, rootComponentFinset_root_mem c hc⟩
have hAB : Disjoint A B := (rootComponentFinset_disjoint_complement D c).symm
have hUnion : A ∪ B = D := by
rw [Finset.union_comm]
exact rootComponentFinset_union_complement D c
have hEdges : inducedEdges (A ∪ B) =
insert (u.1, v.1) (inducedEdges A ∪ inducedEdges B) := by
rw [hUnion, inducedEdges_rootComponent_split hroot c hc]
congr 1
exact Finset.union_comm _ _
have h := field_union_fst beta A B hAB u v hEdges y
have hu := field_congr_domain beta hUnion
(⟨u, Finset.mem_union_left B u.property⟩ : ↥(A ∪ B))
⟨ConnectedHull.root d, hroot⟩ rfl y
rw [hu] at h
exact h
end Problem367.FiniteIsingSplit
end
end OAI
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