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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 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphGeometry.FiniteInduced
import OAI.Probability.CriticalPercolation.GraphGeometry.HitEvent
import OAI.Probability.CriticalPercolation.GraphGeometry.VertexBall
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory ProbabilityTheory
open scoped Classical BigOperators ENNReal
namespace CriticalPercolation.BondGraph
open BernoulliProduct
variable {V E : Type*} (G : BondGraph V E)
def exteriorCluster (π : V → ℤ) (j : ℤ) (o : V) (ω : Set E) : Set V :=
G.cluster (ω∩G.inducedEdges {v | j ≤ π v}) o
def levelContacts (π : V → ℤ) (j : ℤ) (o : V) (ω : Set E) : Set V :=
G.exteriorCluster π j o ω∩{v | π v=j}
def contactDrop (π : V → ℤ) (j : ℤ) (o : V) (ω : Set E) : Set E :=
{e | ∃ v∈G.levelContacts π j o ω, ∃ w, π w<j ∧ G.ends e=s(v,w)}
lemma contactDrop_subset_incident (π : V → ℤ) (j : ℤ) (o : V) (ω : Set E) :
G.contactDrop π j o ω ⊆ G.incidentEdges (G.levelContacts π j o ω) := by
rintro e ⟨v,hv,w,hw,he⟩
exact ⟨v,hv,by rw [he]; simp⟩
lemma contactDrop_disjoint_exterior (π : V → ℤ) (j : ℤ) (o : V) (ω : Set E) :
Disjoint (G.contactDrop π j o ω) (G.inducedEdges {v | j ≤ π v}) := by
apply Set.disjoint_left.mpr
rintro e ⟨v,hv,w,hw,he⟩ he'
exact (not_le.mpr hw) (he' w (by rw [he]; simp))
lemma exteriorCluster_subset_level (π : V → ℤ) (j : ℤ) (o : V)
(ho : j ≤ π o) (ω : Set E) : G.exteriorCluster π j o ω ⊆ {v | j ≤ π v} := by
intro v hv
have hc : ∀ ⦃x y⦄, x∈{v | j ≤ π v} →
(G.openGraph (ω∩G.inducedEdges {v | j ≤ π v})).Adj x y → y∈{v | j ≤ π v} := by
intro x y hx hxy
obtain ⟨⟨e,he,hend⟩,_⟩ := hxy
exact he.2 y (by rw [hend]; simp)
exact ClusterDeletion.reachable_mem_of_closed hc ho hv
lemma cluster_eq_exterior_of_drop_closed (π : V → ℤ) (j : ℤ) (o : V)
(ho : j ≤ π o)
(hcross : ∀ e v w, G.ends e=s(v,w) → j ≤ π v → π w<j → π v=j)
(ω : Set E) (hclosed : Disjoint (G.contactDrop π j o ω) ω) :
G.cluster ω o=G.exteriorCluster π j o ω := by
apply Set.Subset.antisymm
· intro v hv
have hc : ∀ ⦃x y⦄, x∈G.exteriorCluster π j o ω →
(G.openGraph ω).Adj x y → y∈G.exteriorCluster π j o ω := by
intro x y hx hxy
obtain ⟨⟨e,he,hend⟩,hne⟩ := hxy
have hxj := G.exteriorCluster_subset_level π j o ho ω hx
by_cases hy : j ≤ π y
· apply hx.trans
apply SimpleGraph.Adj.reachable
refine ⟨⟨e,⟨he,?_⟩,hend⟩,hne⟩
intro z hz
rw [hend] at hz
rcases Sym2.mem_iff.mp hz with rfl|rfl
· exact hxj
· exact hy
· have hy' := lt_of_not_ge hy
exact False.elim (Set.disjoint_left.mp hclosed
⟨x,⟨hx,hcross e x y hend hxj hy'⟩,y,hy',hend⟩ he)
exact ClusterDeletion.reachable_mem_of_closed hc (SimpleGraph.Reachable.refl o) hv
· exact G.cluster_mono Set.inter_subset_left o
def minimumLevel (π : V → ℤ) (o : V) (j : ℤ) : Set (Set E) :=
G.hitEvent o {v | π v=j}\G.hitEvent o {v | π v<j}
lemma minimumLevel_of_drop_closed (π : V → ℤ) (j : ℤ) (o : V)
(ho : j ≤ π o)
(hcross : ∀ e v w, G.ends e=s(v,w) → j ≤ π v → π w<j → π v=j)
(ω : Set E) (hcontact : (G.levelContacts π j o ω).Nonempty)
(hclosed : Disjoint (G.contactDrop π j o ω) ω) :
ω∈G.minimumLevel π o j := by
have he := G.cluster_eq_exterior_of_drop_closed π j o ho hcross ω hclosed
constructor
· obtain ⟨v,hv,hj⟩ := hcontact
exact ⟨v,he.symm ▸ hv,hj⟩
· rintro ⟨v,hv,hj⟩
have hlt : π v < j := hj
have hle : j ≤ π v := G.exteriorCluster_subset_level π j o ho ω (he ▸ hv)
omega
lemma minimumLevel_pairwiseDisjoint (π : V → ℤ) (o : V) :
Pairwise (fun i j => Disjoint (G.minimumLevel π o i) (G.minimumLevel π o j)) := by
intro i j hij
apply Set.disjoint_left.mpr
rintro ω ⟨⟨v,hv,hvi⟩,hnoti⟩ ⟨⟨w,hw,hwj⟩,hnotj⟩
rcases lt_or_gt_of_ne hij with hlt|hgt
· exact hnotj ⟨v,hv,by change π v < j; rw [hvi]; exact hlt⟩
· exact hnoti ⟨w,hw,by change π w < i; rw [hwj]; exact hgt⟩
lemma measurableSet_minimumLevel [Countable V] [Countable E] (π : V → ℤ) (o : V) (j : ℤ) :
MeasurableSet (G.minimumLevel π o j) :=
(G.measurableSet_hitEvent o _).diff (G.measurableSet_hitEvent o _)
end CriticalPercolation.BondGraph
end
end
end
end OAI
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