| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Exploration.SliceExploration | |
| import OAI.Probability.CriticalPercolation.Gluing.StripMoves | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.MeasurableSetRestrict | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set MeasureTheory ProbabilityTheory Filter | |
| open scoped Classical ENNReal Topology | |
| namespace CriticalPercolation.BondGraph | |
| variable {V E : Type*} (G : BondGraph V E) | |
| def freshExitEvent (S : Set V) (v : V) (N : ℕ) : Set (Set E) := | |
| {ω | ∃ F : Finset E, N≤F.card ∧ | |
| (F : Set E)⊆G.freshClusterBoundary (G.inducedEdges S) v ω∩ω} | |
| lemma freshClusterBoundary_induced_mono {S : Set V} {v : V} (hv : v∈S) | |
| {ω η : Set E} (hωη : ω⊆η) : | |
| G.freshClusterBoundary (G.inducedEdges S) v ω ⊆ | |
| G.freshClusterBoundary (G.inducedEdges S) v η := by | |
| rintro e ⟨⟨x,y,he,hx,hy⟩,heS⟩ | |
| have hyS : y∈S := G.cluster_induced_subset ω S hv hy | |
| have hxS : x∉S := by | |
| intro hxS | |
| apply heS | |
| intro z hz | |
| rw [he] at hz | |
| rcases Sym2.mem_iff.mp hz with rfl|rfl | |
| · exact hxS | |
| · exact hyS | |
| refine ⟨⟨x,y,he,fun hx' => hxS (G.cluster_induced_subset η S hv hx'),?_⟩,heS⟩ | |
| exact G.cluster_mono (Set.inter_subset_inter_left _ hωη) v hy | |
| lemma freshExitEvent_upper {S : Set V} {v : V} (hv : v∈S) (N : ℕ) : | |
| IsUpperSet (G.freshExitEvent S v N) := by | |
| rintro ω η hωη ⟨F,hF,hFω⟩ | |
| exact ⟨F,hF,fun e he => | |
| ⟨G.freshClusterBoundary_induced_mono hv hωη (hFω he).1,hωη (hFω he).2⟩⟩ | |
| lemma freshExitEvent_finiteWitness {S : Set V} {v : V} (hv : v∈S) (N : ℕ) : | |
| BernoulliProduct.HasFiniteOpenWitness (G.freshExitEvent S v N) := by | |
| rintro ω ⟨F,hF,hFω⟩ | |
| let X := G.inducedEdges S | |
| let C := G.cluster (ω∩X) v | |
| let T : Set V := (⋃ e∈(F : Set E), (↑(G.ends e).toFinset : Set V))∩C | |
| have hT : T.Finite := | |
| (F.finite_toSet.biUnion (fun e _ => (G.ends e).toFinset.finite_toSet)).inter_of_left C | |
| obtain ⟨J,hJ,hJω,hconn⟩ := G.finite_subset_cluster_finite_witness hT Set.inter_subset_right | |
| have hsub : J∪(F : Set E)⊆ω := Set.union_subset | |
| (hJω.trans Set.inter_subset_left) (fun e he => (hFω he).2) | |
| refine ⟨J∪(F : Set E),hJ.union F.finite_toSet,hsub,F,hF,fun e he => ?_⟩ | |
| obtain ⟨⟨x,y,hexy,_,hy⟩,heX⟩ := (hFω he).1 | |
| have hyT : y∈T := by | |
| refine ⟨Set.mem_iUnion₂.mpr ⟨e,he,?_⟩,hy⟩ | |
| have hm : y∈G.ends e := by rw [hexy]; simp | |
| simpa only [Finset.mem_coe,Sym2.mem_toFinset] using hm | |
| have hyJ : y∈G.cluster ((J∪(F : Set E))∩X) v := | |
| G.cluster_mono (Set.subset_inter Set.subset_union_left | |
| (hJω.trans Set.inter_subset_right)) v (hconn hyT) | |
| have hx' : x∉G.cluster ((J∪(F : Set E))∩X) v := by | |
| intro hreach | |
| apply heX | |
| intro vertex hvertex | |
| rw [hexy] at hvertex | |
| rcases Sym2.mem_iff.mp hvertex with rfl | rfl | |
| · exact G.cluster_induced_subset (J∪(F : Set E)) S hv hreach | |
| · exact G.cluster_induced_subset ω S hv hy | |
| exact ⟨⟨⟨x,y,hexy,hx',hyJ⟩,heX⟩,Or.inr he⟩ | |
| lemma measurableSet_freshExitEvent [Countable E] {S : Set V} {v : V} | |
| (hv : v∈S) (N : ℕ) : MeasurableSet (G.freshExitEvent S v N) := | |
| BernoulliProduct.measurableSet_finiteWitness _ (G.freshExitEvent_upper hv N) | |
| (G.freshExitEvent_finiteWitness hv N) | |
| lemma freshExitEvent_of_card (S : Set V) (v : V) (N : ℕ) (ω : Set E) | |
| (hf : (G.freshClusterBoundary (G.inducedEdges S) v ω∩ω).Finite) | |
| (hN : N≤(G.freshClusterBoundary (G.inducedEdges S) v ω∩ω).ncard) : | |
| ω∈G.freshExitEvent S v N := by | |
| exact ⟨hf.toFinset,by rwa [← Set.ncard_eq_toFinset_card _ hf], | |
| fun e he => hf.mem_toFinset.mp he⟩ | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end OAI | |