| import OAI.Probability.ClassicalON.BlockLocal |
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| namespace OAI |
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| universe uK |
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| noncomputable section |
| open Set |
| open scoped Classical |
| namespace ClassicalON.LatticeGraph |
| variable {K : Type uK} |
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|
| 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 |
|
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| 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))) |
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| @[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 |
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| 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 |
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| 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 |
| |
