openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Clusters /InfiniteUniqueness.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Criticality.FiniteEnergy | |
| import OAI.Probability.CriticalPercolation.Clusters.InfiniteUniquenessGeometry | |
| import OAI.Probability.CriticalPercolation.Clusters.PercolationTail | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set MeasureTheory ProbabilityTheory | |
| open scoped Classical | |
| namespace CriticalPercolation.BondGraph | |
| variable {V E : Type*} [Countable V] [Countable E] (G : BondGraph V E) | |
| def atMostOneInfinite : Set (Set E) := | |
| {ω | ∀ x y, (G.cluster ω x).Infinite → (G.cluster ω y).Infinite → | |
| (G.openGraph ω).Reachable x y} | |
| lemma measurableSet_atMostOneInfinite : MeasurableSet G.atMostOneInfinite := by | |
| have heq : G.atMostOneInfinite = ⋂ x, ⋂ y, (G.distinctInfiniteEvent x y)ᶜ := by | |
| ext ω | |
| simp only [atMostOneInfinite,distinctInfiniteEvent,Set.mem_ofPred_eq,Set.mem_iInter, | |
| Set.mem_compl_iff] | |
| constructor | |
| · intro h x y hh | |
| exact hh.2.2 (h x y hh.1 hh.2.1) | |
| · intro h x y hx hy | |
| by_contra hr | |
| exact h x y ⟨hx,hy,hr⟩ | |
| rw [heq] | |
| exact MeasurableSet.iInter fun x => MeasurableSet.iInter fun y => | |
| (G.measurableSet_distinctInfiniteEvent x y).compl | |
| theorem atMostOneInfinite_ae (hconnected : G.fullGraph.Connected) | |
| (hlocal : G.LocallyFinite) (hquasi : G.QuasiTransitive) | |
| (hg : G.SubexponentialGrowth) (p : unitInterval) | |
| (hp : (p:ℝ) ∈ Set.Ioo 0 1) : ∀ᵐ ω ∂G.law p, ω ∈ G.atMostOneInfinite := by | |
| have hn : ∀ (S : Finset E) (x y : V), ∀ᵐ ω ∂G.law p, | |
| G.fullGraph.Adj x y → (S : Set E) ∪ ω ∉ G.distinctInfiniteEvent x y := by | |
| intro S x y | |
| by_cases hxy : G.fullGraph.Adj x y | |
| · have hz := BernoulliProduct.finite_union_preimage_null (fun _ : E => p) | |
| (fun _ => hp.1) (S : Set E) S.finite_toSet (G.distinctInfiniteEvent x y) | |
| (G.measurableSet_distinctInfiniteEvent x y) | |
| (by rw [BernoulliProduct.law_const]; | |
| exact G.adjacent_distinctInfinite_null hconnected hlocal hquasi hg p hp x y hxy) | |
| rw [BernoulliProduct.law_const] at hz | |
| have hh : ∀ᵐ ω ∂G.law p, (S : Set E) ∪ ω ∉ G.distinctInfiniteEvent x y := by | |
| rw [ae_iff] | |
| simp only [not_not] | |
| exact hz | |
| exact hh.mono (fun _ h _ => h) | |
| · exact Filter.Eventually.of_forall (fun _ h => False.elim (hxy h)) | |
| have hall := ae_all_iff.mpr (fun S => ae_all_iff.mpr (fun x => ae_all_iff.mpr (hn S x))) | |
| filter_upwards [hall] with ω hω | |
| intro x y hx hy | |
| by_contra hxy | |
| obtain ⟨S,hS,a,b,hab,hh⟩ := G.distinctInfinite_force_adjacent hconnected ω x y ⟨hx,hy,hxy⟩ | |
| have hnot := hω hS.toFinset a b hab | |
| exact hnot (by simpa only [Set.Finite.coe_toFinset] using hh) | |
| theorem uniqueInfinite_ae (hconnected : G.fullGraph.Connected) | |
| (hlocal : G.LocallyFinite) (hquasi : G.QuasiTransitive) | |
| (hg : G.SubexponentialGrowth) (p : unitInterval) | |
| (hp : (p:ℝ) ∈ Set.Ioo 0 1) (hpositive : 0 < G.law p G.percolates) : | |
| ∀ᵐ ω ∂G.law p, ∃ x, (G.cluster ω x).Infinite ∧ | |
| ∀ y, (G.cluster ω y).Infinite → (G.openGraph ω).Reachable x y := by | |
| have hex : ∀ᵐ ω ∂G.law p, ω ∈ G.percolates := by | |
| rw [ae_iff] | |
| change G.law p G.percolatesᶜ = 0 | |
| rw [measure_compl G.measurableSet_percolates (measure_ne_top _ _), | |
| G.percolates_probability_one p hpositive, measure_univ, tsub_self] | |
| filter_upwards [hex,G.atMostOneInfinite_ae hconnected hlocal hquasi hg p hp] with ω hω hu | |
| obtain ⟨x,hx⟩ := hω | |
| exact ⟨x,hx,fun y hy => hu x y hx hy⟩ | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end | |
| end OAI | |