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f205175 | 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 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 | 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
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