File size: 3,216 Bytes
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 | import Mathlib
import OAI.Probability.CriticalPercolation.Model
import OAI.Probability.CriticalPercolation.GraphGeometry.VertexBall
import OAI.Probability.CriticalPercolation.ProductMeasure.TailProduct
namespace OAI
section
section
noncomputable section
open Set MeasureTheory ProbabilityTheory
open scoped Classical
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
lemma cluster_union_subset (ω S : Set E) (o : V) :
G.cluster (ω ∪ S) o ⊆ G.cluster ω o ∪
⋃ e ∈ S, ⋃ v ∈ (G.ends e).toFinset, G.cluster ω v := by
intro z hz
have hr := ((G.openGraph (ω ∪ S)).reachable_iff_reflTransGen o z).mp hz
clear hz
induction hr with
| refl => exact Or.inl (.refl _)
| @tail x y hr hxy ih =>
obtain ⟨⟨e, he, heq⟩,hne⟩ := (SimpleGraph.fromEdgeSet_adj _).mp hxy
rcases he with he | he
· have hadj : (G.openGraph ω).Adj x y :=
(SimpleGraph.fromEdgeSet_adj _).mpr ⟨⟨e,he,heq⟩,hne⟩
rcases ih with ho | h
· exact Or.inl (ho.trans hadj.reachable)
· obtain ⟨f,hf,h⟩ := Set.mem_iUnion₂.mp h
obtain ⟨v,hv,hh⟩ := Set.mem_iUnion₂.mp h
exact Or.inr (Set.mem_iUnion₂.mpr ⟨f,hf,Set.mem_iUnion₂.mpr
⟨v,hv,hh.trans hadj.reachable⟩⟩)
· exact Or.inr (Set.mem_iUnion₂.mpr ⟨e,he,Set.mem_iUnion₂.mpr
⟨y,by simp [heq], .refl _⟩⟩)
lemma percolates_union_finite_iff (ω S : Set E) (hS : S.Finite) :
ω ∪ S ∈ G.percolates ↔ ω ∈ G.percolates := by
constructor
· rintro ⟨o,ho⟩
by_contra hω
have hfinite : ∀ v, (G.cluster ω v).Finite :=
fun v => Set.not_infinite.mp (fun hv => hω ⟨v,hv⟩)
exact ho (((hfinite o).union (hS.biUnion fun e _ =>
(G.ends e).toFinset.finite_toSet.biUnion fun v _ => hfinite v)).subset
(G.cluster_union_subset ω S o))
· exact G.percolates_mono Set.subset_union_left
lemma percolates_diff_finite_iff (ω S : Set E) (hS : S.Finite) :
ω \ S ∈ G.percolates ↔ ω ∈ G.percolates := by
constructor
· exact G.percolates_mono Set.sdiff_subset
· intro hω
have hh := G.percolates_mono (show ω ⊆ (ω \ S) ∪ S from
fun e he => by by_cases hs : e ∈ S; exact Or.inr hs; exact Or.inl ⟨he,hs⟩) hω
exact (G.percolates_union_finite_iff (ω \ S) S hS).mp hh
theorem percolates_zero_one [Countable V] [Countable E] (q : E → unitInterval) :
BernoulliProduct.law q G.percolates = 0 ∨
BernoulliProduct.law q G.percolates = 1 := by
apply BernoulliProduct.zero_one_of_finite_erasure_invariant q G.percolates
G.measurableSet_percolates
intro S hS ω
have heq : BernoulliProduct.forceOn S ∅ ω = ω \ S := by
ext e
change (if e ∈ S then False else e ∈ ω) ↔ (e ∈ ω ∧ e ∉ S)
split_ifs <;> tauto
rw [heq]
exact G.percolates_diff_finite_iff ω S hS
lemma percolates_probability_one [Countable V] [Countable E] (p : unitInterval)
(h : 0 < G.law p G.percolates) : G.law p G.percolates = 1 := by
have hh := G.percolates_zero_one (fun _ => p)
rw [BernoulliProduct.law_const] at hh
exact hh.resolve_left (ne_of_gt h)
end CriticalPercolation.BondGraph
end
end
end
end OAI
|