| import OAI.Probability.ClassicalON.SeparatedAnnuli |
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| namespace OAI |
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| noncomputable section |
| open Set |
| open scoped Classical |
| namespace ClassicalON.LatticeGraph |
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| def extendAnnulusSign (G : LatticeGraph) (N : ℤ) (z : Site) |
| (s : (G.annulusGraph N z).vertices → Bool) (v : G.vertices) : Bool := |
| if h : v.val∈annulusSites N z then |
| s ⟨v.val,Finset.mem_filter.mpr ⟨v.property,h⟩⟩ else decide (2*N<siteRadius z v.val) |
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| theorem extendAnnulusSign_in (G : LatticeGraph) (N : ℤ) (z : Site) |
| (s : (G.annulusGraph N z).vertices → Bool) (v : (G.annulusGraph N z).vertices) : |
| G.extendAnnulusSign N z s (G.restrictVertex _ v)=s v := by |
| simp only [extendAnnulusSign,show (G.restrictVertex _ v).val∈annulusSites N z from |
| (Finset.mem_filter.mp v.property).2,↓reduceDIte] |
| rfl |
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| theorem extendAnnulusSign_inner (G : LatticeGraph) (N : ℤ) (hN : 0<N) (z : Site) |
| (s : (G.annulusGraph N z).vertices → Bool) |
| (hs : ∀ v∈G.annulusPins N z,s v=layerSigns (G.annulusInner N z) v) |
| (v : G.vertices) (hv : siteRadius z v.val≤N) : G.extendAnnulusSign N z s v=false := by |
| unfold extendAnnulusSign |
| split_ifs with h |
| · let w : (G.annulusGraph N z).vertices := ⟨v.val,Finset.mem_filter.mpr ⟨v.property,h⟩⟩ |
| have hw : w∈G.annulusInner N z := le_antisymm hv h.1 |
| have hh := hs w (Or.inl hw) |
| simpa only [layerSigns,hw,ite_true] using hh |
| · have hh : ¬2*N<siteRadius z v.val := by omega |
| simp only [decide_eq_false hh] |
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| theorem extendAnnulusSign_outer (G : LatticeGraph) (N : ℤ) (hN : 0<N) (z : Site) |
| (s : (G.annulusGraph N z).vertices → Bool) |
| (hs : ∀ v∈G.annulusPins N z,s v=layerSigns (G.annulusInner N z) v) |
| (v : G.vertices) (hv : 2*N≤ siteRadius z v.val) : G.extendAnnulusSign N z s v=true := by |
| unfold extendAnnulusSign |
| split_ifs with h |
| · let w : (G.annulusGraph N z).vertices := ⟨v.val,Finset.mem_filter.mpr ⟨v.property,h⟩⟩ |
| have hw : w∈G.annulusOuter N z := le_antisymm h.2 hv |
| have hw' : w∉G.annulusInner N z := by change siteRadius z v.val≠N; omega |
| have hh := hs w (Or.inr hw) |
| simpa only [layerSigns,hw',ite_false] using hh |
| · have hh : 2*N<siteRadius z v.val := by |
| by_contra hn |
| apply h |
| change N≤ siteRadius z v.val ∧ siteRadius z v.val≤2*N |
| omega |
| simp only [decide_eq_true hh] |
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| theorem extendAnnulusSign_compatible (G : LatticeGraph) (N : ℤ) (hN : 0<N) (z : Site) |
| (η : G.edges → Bool) (s : (G.annulusGraph N z).vertices → Bool) |
| (hs : s∈prescribedCompatibleSigns (fun e : (G.annulusGraph N z).edges => e.val.1) |
| (fun e => e.val.2) (G.annulusPins N z) (layerSigns (G.annulusInner N z)) |
| (fun e => η (G.restrictEdge _ e))) : |
| G.extendAnnulusSign N z s∈compatibleSigns (fun e : G.edges => e.val.1) (fun e => e.val.2) η := by |
| intro e he |
| have hstep := siteRadius_neighbor_step z (Or.inl (G.nearest e.val e.property)) |
| by_cases hl : siteRadius z e.val.1.val<N |
| · rw [G.extendAnnulusSign_inner N hN z s hs.2 e.val.1 (by omega), |
| G.extendAnnulusSign_inner N hN z s hs.2 e.val.2 (by omega)] |
| by_cases hl' : 2*N<siteRadius z e.val.1.val |
| · rw [G.extendAnnulusSign_outer N hN z s hs.2 e.val.1 (by omega), |
| G.extendAnnulusSign_outer N hN z s hs.2 e.val.2 (by omega)] |
| by_cases hr : siteRadius z e.val.2.val<N |
| · rw [G.extendAnnulusSign_inner N hN z s hs.2 e.val.1 (by omega), |
| G.extendAnnulusSign_inner N hN z s hs.2 e.val.2 (by omega)] |
| by_cases hr' : 2*N<siteRadius z e.val.2.val |
| · rw [G.extendAnnulusSign_outer N hN z s hs.2 e.val.1 (by omega), |
| G.extendAnnulusSign_outer N hN z s hs.2 e.val.2 (by omega)] |
| have h1 : e.val.1.val∈annulusSites N z := ⟨by omega,by omega⟩ |
| have h2 : e.val.2.val∈annulusSites N z := ⟨by omega,by omega⟩ |
| let e' := G.restrictEdgeEquiv (·∈annulusSites N z) ⟨e,h1,h2⟩ |
| have hcomp := hs.1 e' he |
| rw [← G.extendAnnulusSign_in N z s e'.val.1,← G.extendAnnulusSign_in N z s e'.val.2] at hcomp |
| exact hcomp |
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|
| theorem annulus_crossing_of_connected (G : LatticeGraph) (N : ℤ) (hN : 0<N) (z : Site) |
| (η : G.edges → Bool) (x y : G.vertices) (hx : siteRadius z x.val≤N) (hy : 2*N≤ siteRadius z y.val) |
| (hxy : bondConnected (fun e : G.edges => e.val.1) (fun e => e.val.2) η x y) : |
| G.annulusIndicator N z η=1 := by |
| have hd : Disjoint (G.annulusInner N z) (G.annulusOuter N z) := by |
| apply Set.disjoint_left.mpr |
| intro v hv hw |
| change siteRadius z v.val=N at hv |
| change siteRadius z v.val=2*N at hw |
| omega |
| have hc : bondCrossing (fun e : (G.annulusGraph N z).edges => e.val.1) (fun e => e.val.2) |
| (G.annulusInner N z) (G.annulusOuter N z) (fun e => η (G.restrictEdge _ e)) := by |
| by_contra hn |
| obtain ⟨s,hs⟩ := (layerSigns_consistent_iff (fun e : (G.annulusGraph N z).edges => e.val.1) |
| (fun e => e.val.2) _ _ hd _).mpr hn |
| have he := compatibleSigns_connected _ _ η (G.extendAnnulusSign_compatible N hN z η s hs) hxy |
| rw [G.extendAnnulusSign_inner N hN z s hs.2 x hx,G.extendAnnulusSign_outer N hN z s hs.2 y hy] at he |
| cases he |
| exact ite_eq_left hc |
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| theorem annulus_crossing_of_component (G : LatticeGraph) (N : ℤ) (hN : 0<N) |
| (η : G.edges → Bool) (x y w : G.vertices) (hfar : 4*N≤ siteRadius x.val y.val) |
| (hxy : bondConnected (fun e : G.edges => e.val.1) (fun e => e.val.2) η x y) |
| (hxw : bondConnected (fun e : G.edges => e.val.1) (fun e => e.val.2) η x w) |
| (z : Site) (hw : siteRadius z w.val≤N) : G.annulusIndicator N z η=1 := by |
| have hout : 2*N≤ siteRadius z x.val ∨ 2*N≤ siteRadius z y.val := by |
| have ht := siteRadius_triangle x.val z y.val |
| rw [siteRadius_symm x.val z] at ht |
| omega |
| rcases hout with hx | hy |
| · exact G.annulus_crossing_of_connected N hN z η w x hw hx (bondConnected_symm _ _ _ hxw) |
| · exact G.annulus_crossing_of_connected N hN z η w y hw hy |
| (bondConnected_trans _ _ _ (bondConnected_symm _ _ _ hxw) hxy) |
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| end ClassicalON.LatticeGraph |
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| end |
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| end OAI |
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