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9b6e014 | 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 | import OAI.Probability.ClassicalON.BlockLocal
namespace OAI
universe uK
noncomputable section
open Set
open scoped Classical
namespace ClassicalON.LatticeGraph
variable {K : Type uK}
def annulusFreeEquiv (G : LatticeGraph) (N : ℤ) (hN : 0≤N) (z : K → Site)
(hz : SeparatedCenters N z) (i : K) :
BlockVertex (G.allAnnulusPins N z) (G.annulusVertexLabel N z) (some i) ≃
↥((G.annulusPins N (z i))ᶜ) where
toFun v := ⟨⟨v.val.val.val,Finset.mem_filter.mpr ⟨v.val.val.property,
(centerLabel_some_iff N z hz _ i).mp v.property⟩⟩,
fun hp => v.val.property ⟨i,hp⟩⟩
invFun v := ⟨⟨G.restrictVertex _ v.val,fun hp => v.property
((local_centerPins_iff N hN z hz _ i (Finset.mem_filter.mp v.val.property).2).mp hp)⟩,
(centerLabel_some_iff N z hz _ i).mpr (Finset.mem_filter.mp v.val.property).2⟩
left_inv _ := rfl
right_inv _ := rfl
def annulusEdgesEquiv (G : LatticeGraph) (N : ℤ) (z : K → Site)
(hz : SeparatedCenters N z) (i : K) :
{e : G.edges // G.annulusEdgeLabel N z e=some i} ≃ (G.annulusGraph N (z i)).edges :=
(Equiv.subtypeEquivRight (fun e => G.annulusEdgeLabel_some_iff N z hz e i)).trans
(G.restrictEdgeEquiv (·∈annulusSites N (z i)))
@[simp] theorem annulusEdgesEquiv_restrictEdge (G : LatticeGraph) (N : ℤ) (z : K → Site)
(hz : SeparatedCenters N z) (i : K) (e : {e : G.edges // G.annulusEdgeLabel N z e=some i}) :
G.restrictEdge _ (G.annulusEdgesEquiv N z hz i e)=e.val := rfl
theorem annulus_blockSpin (G : LatticeGraph) (N : ℤ) (hN : 0≤N) (z : K → Site)
(hz : SeparatedCenters N z) (i : K) (s : ↥((G.annulusPins N (z i))ᶜ) → Spin 3)
(v : (G.annulusGraph N (z i)).vertices) :
glueFamily (G.annulusPins N (z i)) (fun _ => poleSpin true) s v=
blockSpin (G.allAnnulusPins N z) (G.annulusVertexLabel N z) (some i)
(fun w => s (G.annulusFreeEquiv N hN z hz i w)) (G.restrictVertex _ v) := by
by_cases hp : v∈G.annulusPins N (z i)
· have hB : G.restrictVertex _ v∈G.allAnnulusPins N z := ⟨i,hp⟩
rw [show glueFamily (G.annulusPins N (z i)) (fun _ => poleSpin true) s v=poleSpin true from
glueFamily_in _ _ _ ⟨v,hp⟩]
simp only [blockSpin,hB,↓reduceDIte]
· have hB : G.restrictVertex _ v∉G.allAnnulusPins N z := by
intro h
exact hp ((local_centerPins_iff N hN z hz v.val i (Finset.mem_filter.mp v.property).2).mp h)
have hl : G.annulusVertexLabel N z (G.restrictVertex _ v)=some i :=
(centerLabel_some_iff N z hz v.val i).mpr (Finset.mem_filter.mp v.property).2
rw [show glueFamily (G.annulusPins N (z i)) (fun _ => poleSpin true) s v=s ⟨v,hp⟩ from
glueFamily_out _ _ _ ⟨v,hp⟩]
simp only [blockSpin,hB,hl,↓reduceDIte]
rfl
theorem annulus_blockMean (G : LatticeGraph) (b : G.edges → ℝ) (N : ℤ) (hN : 0≤N)
(z : K → Site) (hz : SeparatedCenters N z) (i : K)
(f : ((G.annulusGraph N (z i)).edges → Bool) → ℝ) :
poleBondMean (fun e : (G.annulusGraph N (z i)).edges => e.val.1) (fun e => e.val.2)
(G.annulusCoupling b N (z i)) (G.annulusPins N (z i)) (fun _ => true) f=
blockBondMean (fun e : G.edges => e.val.1) (fun e => e.val.2) b
(G.allAnnulusPins N z) (G.annulusVertexLabel N z) (G.annulusEdgeLabel N z) (some i)
(fun η => f (arrowEquiv (G.annulusEdgesEquiv N z hz i) η)) := by
apply blockBondMean_local (fun e : G.edges => e.val.1) (fun e => e.val.2) b
(G.allAnnulusPins N z) (G.annulusVertexLabel N z) (G.annulusEdgeLabel N z) (some i)
(fun e : (G.annulusGraph N (z i)).edges => e.val.1) (fun e => e.val.2)
(G.annulusCoupling b N (z i)) (G.annulusPins N (z i)) (G.annulusFreeEquiv N hN z hz i)
(G.annulusEdgesEquiv N z hz i)
· intro e; rfl
· intro s e
exact G.annulus_blockSpin N hN z hz i s (G.annulusEdgesEquiv N z hz i e).val.1
· intro s e
exact G.annulus_blockSpin N hN z hz i s (G.annulusEdgesEquiv N z hz i e).val.2
end ClassicalON.LatticeGraph
end
end OAI
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