| import Mathlib |
| import OAI.Probability.CriticalPercolation.GraphGeometry.HitEvent |
| import OAI.Probability.CriticalPercolation.Criticality.FiniteEnergy |
| import OAI.Probability.CriticalPercolation.ProductMeasure.ProductEmbedding |
| import OAI.Probability.CriticalPercolation.Gluing.SourceGraph |
| import OAI.Probability.CriticalPercolation.Gluing.SourceLift |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open Set Function MeasureTheory ProbabilityTheory |
| open scoped Classical ENNReal |
|
|
| namespace CriticalPercolation.BondGraph |
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| lemma withSource_reachable_from_seed (A : Set V) (ω : Set (E⊕A)) |
| (hw : ∀ a : A, Sum.inr a∈ω) {v : V} |
| (hv : ∃ a∈A, (G.openGraph (Sum.inl ⁻¹' ω)).Reachable a v) : |
| ((G.withSource A).openGraph ω).Reachable none (some v) := by |
| obtain ⟨a,ha,hav⟩ := hv |
| have hh : ((G.withSource A).openGraph ω).Adj none (some a) := |
| ⟨⟨Sum.inr ⟨a,ha⟩,hw ⟨a,ha⟩,rfl⟩,by simp⟩ |
| exact hh.reachable.trans (G.withSource_original_reachable A ω hav) |
|
|
| lemma withSource_reachable_to_seed (A : Set V) (ω : Set (E⊕A)) {v : V} |
| (hv : ((G.withSource A).openGraph ω).Reachable none (some v)) : |
| ∃ a∈A, (G.openGraph (Sum.inl ⁻¹' ω)).Reachable a v := by |
| have hi : ∀ w, ((G.withSource A).openGraph ω).Reachable none w → |
| Option.elim' True (fun v => ∃ a∈A, (G.openGraph (Sum.inl ⁻¹' ω)).Reachable a v) w := by |
| intro w hw |
| rw [SimpleGraph.reachable_iff_reflTransGen] at hw |
| induction hw with |
| | refl => trivial |
| | @tail u w hr huw ih => |
| obtain ⟨⟨b,hb,hend⟩,hne⟩ := huw |
| cases w with |
| | none => trivial |
| | some w => |
| rcases G.withSource_edge_cases A b u (some w) hend with |
| ⟨b,v,w',rfl,rfl,hw',hb'⟩|⟨a,rfl,hh⟩ |
| · have he : w'=w := Option.some.inj hw'.symm |
| subst w' |
| obtain ⟨a,ha,hav⟩ := ih |
| refine ⟨a,ha,hav.trans ?_⟩ |
| exact (show (G.openGraph (Sum.inl ⁻¹' ω)).Adj v w from |
| ⟨⟨b,hb,hb'⟩,fun h => hne (congrArg some h)⟩).reachable |
| · rcases hh with ⟨rfl,hh⟩|⟨hh,_⟩ |
| · have he : w=a.val := Option.some.inj hh |
| exact ⟨a.val,a.property,he.symm ▸ SimpleGraph.Reachable.refl _⟩ |
| · simp at hh |
| exact hi (some v) hv |
|
|
| lemma source_hit_iff_seedHit (A T : Set V) (ω : Set (E⊕A)) |
| (hw : ∀ a : A, Sum.inr a∈ω) : |
| ω∈(G.withSource A).hitEvent none (some '' T) ↔ |
| Sum.inl ⁻¹' ω∈G.seedHitEvent A T := by |
| constructor |
| · rintro ⟨w,hr,v,hv,rfl⟩ |
| obtain ⟨a,ha,hav⟩ := G.withSource_reachable_to_seed A ω hr |
| exact Set.mem_iUnion₂.mpr ⟨a,ha,v,hav,hv⟩ |
| · intro h |
| obtain ⟨a,ha,v,hav,hv⟩ := Set.mem_iUnion₂.mp h |
| exact ⟨some v,G.withSource_reachable_from_seed A ω hw ⟨a,ha,hav⟩,v,hv,rfl⟩ |
|
|
| lemma source_hit_probability [Countable V] [Countable E] (A T : Set V) |
| (q : (E⊕A) → unitInterval) (hq : ∀ a : A, q (Sum.inr a)=1) : |
| BernoulliProduct.law q ((G.withSource A).hitEvent none (some '' T))= |
| BernoulliProduct.law (fun b => q (Sum.inl b)) (G.seedHitEvent A T) := by |
| have ha (a : A) : ∀ᵐ ω ∂BernoulliProduct.law q, Sum.inr a∈ω := by |
| rw [ae_iff] |
| have he : {ω : Set (E⊕A) | ¬Sum.inr a∈ω}={ω | (Sum.inr a∈ω)=False} := by |
| ext ω; simp |
| rw [he,BernoulliProduct.law_coordinate,hq a] |
| simp [BernoulliProduct.bitMeasure] |
| have hall : ∀ᵐ ω ∂BernoulliProduct.law q, ∀ a : A, Sum.inr a∈ω := ae_all_iff.mpr ha |
| have heq : BernoulliProduct.law q ((G.withSource A).hitEvent none (some '' T))= |
| BernoulliProduct.law q ((fun ω : Set (E⊕A) => Sum.inl ⁻¹' ω) ⁻¹' G.seedHitEvent A T) := by |
| apply measure_congr |
| filter_upwards [hall] with ω hω |
| exact propext (G.source_hit_iff_seedHit A T ω hω) |
| rw [heq] |
| change BernoulliProduct.law q ((fun ω : Set (E⊕A) => fun b => Sum.inl b∈ω) ⁻¹' G.seedHitEvent A T)=_ |
| rw [← Measure.map_apply |
| (Measurable.of_eval (fun bond => measurable_set_mem (Sum.inl bond))) |
| (G.measurableSet_seedHitEvent A T)] |
| change (Measure.map (fun ω : Set (E⊕A) => Sum.inl ⁻¹' ω) (BernoulliProduct.law q)) (G.seedHitEvent A T)=_ |
| rw [BernoulliProduct.law_map_preimage q Sum.inl Sum.inl_injective] |
| rfl |
|
|
| end CriticalPercolation.BondGraph |
| end |
| end |
| end |
|
|
| end OAI |
|
|
