openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Clusters /CriticalClusterMultiplicity.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Criticality.SubcriticalLocalRateBound | |
| import OAI.Probability.CriticalPercolation.Clusters.ClusterNumber | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set MeasureTheory | |
| open scoped Classical | |
| namespace CriticalPercolation.BondGraph | |
| variable {V E : Type*} (G : BondGraph V E) | |
| lemma measurable_infinitelyManyInfiniteClusters [Countable V] [Countable E] : | |
| Measurable (fun ω => G.infinitelyManyInfiniteClusters ω) := by | |
| simp_rw [G.infinitelyManyInfiniteClusters_iff_no_finite_cover] | |
| exact .forall fun S => (G.measurable_infiniteClusterCover (S : Set V)).not | |
| lemma infinitelyManyInfiniteClusters.exists_pairwise {ω : Set E} | |
| (h : G.infinitelyManyInfiniteClusters ω) (n : ℕ) : | |
| ∃ f : Fin n → V, (∀ i, (G.cluster ω (f i)).Infinite) ∧ | |
| Pairwise (fun i j => ¬ (G.openGraph ω).Reachable (f i) (f j)) := by | |
| induction n with | |
| | zero => exact ⟨Fin.elim0,fun i => Fin.elim0 i,fun i => Fin.elim0 i⟩ | |
| | succ n ih => | |
| obtain ⟨f,hf,hpair⟩ := ih | |
| obtain ⟨v,hv,hnew⟩ := h (Finset.univ.image f) | |
| refine ⟨Fin.cons v f,?_,?_⟩ | |
| · intro i | |
| refine Fin.cases hv (fun j => ?_) i | |
| exact hf j | |
| · intro i j hij | |
| revert hij | |
| refine Fin.cases ?_ (fun a => ?_) i | |
| · refine Fin.cases ?_ (fun b hij => ?_) j | |
| · exact fun hij => False.elim (hij rfl) | |
| · exact hnew (f b) (Finset.mem_image.mpr ⟨b,Finset.mem_univ _,rfl⟩) | |
| · refine Fin.cases ?_ (fun b hij => ?_) j | |
| · exact fun _ hab => hnew (f a) (Finset.mem_image.mpr ⟨a,Finset.mem_univ _,rfl⟩) hab.symm | |
| · exact hpair (fun he => hij (congrArg Fin.succ he)) | |
| theorem exponential_critical_infinitelyMany_ae [Infinite V] | |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive) | |
| (o : V) (γ : ℝ) (hγ : 0 < γ) (N : ℕ) | |
| (hgrowth : ∀ m≥N, ∀ v : V, γ*m ≤ Real.log ((G.vertexBall v m).ncard : ℝ)) | |
| (hper : G.law G.criticalProbability G.percolates ≠ 0) : | |
| ∀ᵐ ω ∂G.law G.criticalProbability, G.infinitelyManyInfiniteClusters ω := by | |
| let := G.countable_vertices hc hl | |
| let := G.countable_bonds hc hl | |
| have hp : (0 : ℝ) < G.criticalProbability := by | |
| have hpc : 0 < G.criticalProbability := by | |
| by_contra h | |
| have he : G.criticalProbability = 0 := le_antisymm (not_lt.mp h) bot_le | |
| exact hper (he ▸ G.law_zero_percolates) | |
| exact hpc | |
| rcases G.unique_or_infinitely_many_ae hc hl hq G.criticalProbability hp with hu | hm | |
| · exact False.elim (hper (Sharpness.critical_null_of_unique G hc hl hq o γ hγ N hgrowth hu)) | |
| · exact hm | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end OAI | |