| import Mathlib |
| import OAI.Probability.CriticalPercolation.GraphGeometry.CutEdges |
| import OAI.Probability.CriticalPercolation.GraphGeometry.FiniteInduced |
| import OAI.Probability.CriticalPercolation.GraphGeometry.UniformGeometry |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open Set MeasureTheory ProbabilityTheory Filter |
| open scoped Classical |
|
|
| namespace CriticalPercolation.BondGraph |
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| def heightCut (π : V → ℤ) (t : ℤ) : Set E := G.cutEdges {v | t < π v} |
|
|
| lemma finite_height_interval (π : V → ℤ) (hπ : ∀ t, (π ⁻¹' {t}).Finite) |
| (a b : ℤ) : (π ⁻¹' Set.Icc a b).Finite := by |
| have hh := (Set.finite_Icc a b).biUnion (fun t _ => hπ t) |
| convert hh using 1 |
| ext v |
| simp |
|
|
| lemma finite_heightCut (hlocal : G.LocallyFinite) (π : V → ℤ) |
| (hπ : ∀ t, (π ⁻¹' {t}).Finite) (J : ℕ) |
| (hJ : ∀ e x y, G.ends e=s(x,y) → |π x-π y|≤(J:ℤ)) (t : ℤ) : |
| (G.heightCut π t).Finite := by |
| apply (G.finite_incident hlocal (finite_height_interval π hπ (t-J) t)).subset |
| rintro e ⟨x,y,he,hx,hy⟩ |
| have hxy := (abs_le.mp (hJ e x y he)).1 |
| refine ⟨x,?_,?_⟩ |
| · change t-(J:ℤ) ≤ π x ∧ π x ≤ t |
| change ¬t<π x at hx |
| change t<π y at hy |
| omega |
| · simp [he] |
|
|
| lemma heightCut_disjoint (π : V → ℤ) (J : ℕ) |
| (hJ : ∀ e x y, G.ends e=s(x,y) → |π x-π y|≤(J:ℤ)) |
| (t u : ℤ) (htu : (J:ℤ) ≤ |t-u|) : |
| Disjoint (G.heightCut π t) (G.heightCut π u) := by |
| apply Set.disjoint_left.mpr |
| rintro e ⟨x,y,he,hx,hy⟩ ⟨a,b,hef,ha,hb⟩ |
| change ¬t<π x at hx |
| change t<π y at hy |
| change ¬u<π a at ha |
| change u<π b at hb |
| have hj := abs_le.mp (hJ e x y he) |
| rcases Sym2.eq_iff.mp (he.symm.trans hef) with ⟨hxa,hyb⟩|⟨hxb,hya⟩ |
| · subst a; subst b |
| rcases le_total t u with hl|hl |
| · rw [abs_of_nonpos (sub_nonpos.mpr hl)] at htu |
| omega |
| · rw [abs_of_nonneg (sub_nonneg.mpr hl)] at htu |
| omega |
| · subst b; subst a |
| omega |
|
|
| lemma heightCut_translate (π : V → ℤ) (φ : G.Aut) (a : ℤ) |
| (hφ : ∀ v, π (φ.vertex v)=π v+a) (t : ℤ) : |
| φ.edge '' G.heightCut π t = G.heightCut π (t+a) := by |
| ext e |
| constructor |
| · rintro ⟨f,⟨x,y,hf,hx,hy⟩,rfl⟩ |
| refine ⟨φ.vertex x,φ.vertex y,?_,?_,?_⟩ |
| · rw [φ.preserves,hf,Sym2.map_mk] |
| · change ¬t+a<π (φ.vertex x) |
| rw [hφ] |
| change ¬t<π x at hx |
| omega |
| · change t+a<π (φ.vertex y) |
| rw [hφ] |
| change t<π y at hy |
| omega |
| · rintro ⟨x,y,he,hx,hy⟩ |
| refine ⟨φ.edge.symm e,?_,φ.edge.apply_symm_apply e⟩ |
| refine ⟨φ.vertex.symm x,φ.vertex.symm y,?_,?_,?_⟩ |
| · have hh := (Aut.symm G φ).preserves e |
| change G.ends (φ.edge.symm e) = s(φ.vertex.symm x,φ.vertex.symm y) |
| change G.ends (φ.edge.symm e) = Sym2.map φ.vertex.symm (G.ends e) at hh |
| rw [hh,he,Sym2.map_mk] |
| · have hh := hφ (φ.vertex.symm x) |
| simp only [Equiv.apply_symm_apply] at hh |
| change ¬t<π (φ.vertex.symm x) |
| change ¬t+a<π x at hx |
| omega |
| · have hh := hφ (φ.vertex.symm y) |
| simp only [Equiv.apply_symm_apply] at hh |
| change t<π (φ.vertex.symm y) |
| change t+a<π y at hy |
| omega |
|
|
| lemma heightCut_ncard (π : V → ℤ) |
| (htrans : ∀ t, ∃ φ : G.Aut, ∀ v, π (φ.vertex v)=π v+t) (t : ℤ) : |
| (G.heightCut π t).ncard = (G.heightCut π 0).ncard := by |
| obtain ⟨φ,hφ⟩ := htrans t |
| have he := G.heightCut_translate π φ t hφ 0 |
| simp only [zero_add] at he |
| rw [← he,Set.ncard_image_of_injective _ φ.edge.injective] |
|
|
| lemma cluster_bounded_by_closed_cuts (π : V → ℤ) (ω : Set E) (a b : ℤ) |
| (ha : Disjoint (G.heightCut π a) ω) (hb : Disjoint (G.heightCut π b) ω) |
| (o : V) (hoa : a<π o) (hob : π o≤b) : |
| G.cluster ω o ⊆ π ⁻¹' Set.Icc (a+1) b := by |
| intro v hv |
| constructor |
| · by_contra h |
| have hva : ¬a<π v := by change ¬a+1≤π v at h; omega |
| exact G.not_reachable_of_cut_closed {v | a<π v} ω |
| (fun e he => Set.disjoint_left.mp ha he) hva hoa hv.symm |
| · by_contra h |
| have hvb : b<π v := lt_of_not_ge h |
| exact G.not_reachable_of_cut_closed {v | b<π v} ω |
| (fun e he => Set.disjoint_left.mp hb he) (not_lt.mpr hob) hvb hv |
|
|
| end CriticalPercolation.BondGraph |
| end |
| end |
| end |
|
|
| end OAI |
|
|
