| 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 |
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|
