openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /GraphGeometry /InfiniteEventSubsetHit.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.HitEvent | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.OpenGraphIso | |
| namespace OAI | |
| section | |
| section | |
| open scoped BigOperators | |
| open scoped Classical ENNReal | |
| open Set | |
| open Set MeasureTheory ProbabilityTheory Filter | |
| open scoped Classical Topology | |
| open Set MeasureTheory ProbabilityTheory | |
| open scoped Classical | |
| open Filter Topology | |
| namespace CriticalPercolation.BondGraph | |
| variable {V E : Type*} (G : BondGraph V E) | |
| lemma infiniteEvent_subset_hit_compl (o : V) (K : Set V) (hK : K.Finite) : | |
| G.infiniteEvent o ⊆ G.hitEvent o Kᶜ := by | |
| intro ω hω | |
| exact (hω.sdiff hK).nonempty | |
| lemma infiniteEvent_eq_iInter_hit_compl | |
| (K : Set.FiniteExhaustion (Set.univ : Set V)) (o : V) : | |
| G.infiniteEvent o = ⋂ n, G.hitEvent o (K n)ᶜ := by | |
| ext ω | |
| constructor | |
| · intro hω | |
| exact Set.mem_iInter.mpr fun n => G.infiniteEvent_subset_hit_compl o (K n) (K.finite n) hω | |
| · intro hω | |
| change (G.cluster ω o).Infinite | |
| intro hfin | |
| obtain ⟨n, hn⟩ := finite_subset_exhaustion K hfin (Set.subset_univ _) | |
| obtain ⟨v, hv, hvK⟩ := Set.mem_iInter.mp hω n | |
| exact hvK (hn hv) | |
| lemma tendsto_real_hit_compl [Countable V] [Countable E] | |
| (p : unitInterval) (K : Set.FiniteExhaustion (Set.univ : Set V)) (o : V) : | |
| Tendsto (fun n => (G.law p).real (G.hitEvent o (K n)ᶜ)) atTop | |
| (𝓝 ((G.law p).real (G.infiniteEvent o))) := by | |
| have h := tendsto_measure_iInter_atTop (μ := G.law p) | |
| (fun n => (G.measurableSet_hitEvent o (K n)ᶜ).nullMeasurableSet) | |
| (show Antitone (fun n => G.hitEvent o (K n)ᶜ) from | |
| fun _ _ h => G.hitEvent_mono_target o (Set.compl_subset_compl.mpr (K.mono h))) | |
| ⟨0, measure_ne_top _ _⟩ | |
| rw [← G.infiniteEvent_eq_iInter_hit_compl K o] at h | |
| exact (ENNReal.tendsto_toReal (measure_ne_top _ _)).comp h | |
| lemma real_hit_singleton_pos (hconnected : G.fullGraph.Connected) | |
| (p : unitInterval) (hp : 0 < (p : ℝ)) (o v : V) : | |
| 0 < (G.law p).real (G.hitEvent o {v}) := by | |
| obtain ⟨s, hs, _, hsv⟩ := G.mem_cluster_finite_witness (hconnected o v) | |
| have hsub : {ω : Set E | s ⊆ ω} ⊆ G.hitEvent o {v} := by | |
| intro ω hω | |
| exact ⟨v, G.cluster_mono hω o hsv, rfl⟩ | |
| have hpos : 0 < (G.law p).real {ω : Set E | s ⊆ ω} := by | |
| rw [law, ← BernoulliProduct.law_const (E := E) p] | |
| rw [BernoulliProduct.real_law_subset p s hs] | |
| exact pow_pos hp _ | |
| exact lt_of_lt_of_le hpos (measureReal_mono hsub) | |
| lemma real_infiniteEvent_transfer [Countable V] [Countable E] | |
| (p : unitInterval) (o v : V) : | |
| (G.law p).real (G.hitEvent o {v}) * (G.law p).real (G.infiniteEvent v) ≤ | |
| (G.law p).real (G.infiniteEvent o) := by | |
| let K := (Set.to_countable (Set.univ : Set V)).finiteExhaustion | |
| have hn (n : ℕ) : | |
| (G.law p).real (G.hitEvent o {v}) * (G.law p).real (G.infiniteEvent v) ≤ | |
| (G.law p).real (G.hitEvent o (K n)ᶜ) := by | |
| apply le_trans (G.joint_gluing p o {v} (K n)ᶜ | |
| ((G.law p).real (G.infiniteEvent v)) ?_) | |
| (measureReal_mono Set.inter_subset_right) | |
| intro w hw | |
| rcases Set.mem_singleton_iff.mp hw with rfl | |
| exact measureReal_mono (G.infiniteEvent_subset_hit_compl w (K n) (K.finite n)) | |
| exact ge_of_tendsto (G.tendsto_real_hit_compl p K o) (Eventually.of_forall hn) | |
| lemma real_infiniteEvent_pos_all [Countable V] [Countable E] | |
| (hconnected : G.fullGraph.Connected) (p : unitInterval) (hp : 0 < (p : ℝ)) | |
| {v : V} (hv : 0 < (G.law p).real (G.infiniteEvent v)) (o : V) : | |
| 0 < (G.law p).real (G.infiniteEvent o) := | |
| lt_of_lt_of_le (mul_pos (G.real_hit_singleton_pos hconnected p hp o v) hv) | |
| (G.real_infiniteEvent_transfer p o v) | |
| theorem uniform_rooted_probability | |
| (hconnected : G.fullGraph.Connected) (hlocal : G.LocallyFinite) | |
| (hquasi : G.QuasiTransitive) (p : unitInterval) | |
| (hp : 0 < G.law p G.percolates) : | |
| ∃ θ : ℝ, 0 < θ ∧ ∀ o : V, θ ≤ (G.law p).real (G.infiniteEvent o) := by | |
| let := G.countable_vertices hconnected hlocal | |
| let := G.countable_bonds hconnected hlocal | |
| have hp0 : 0 < (p : ℝ) := by | |
| by_contra h | |
| have hz : p = 0 := Subtype.ext (le_antisymm (not_lt.mp h) p.property.1) | |
| subst p | |
| rw [G.law_zero_percolates] at hp | |
| exact (lt_irrefl _ hp) | |
| have hn : ¬ ∀ v, G.law p (G.infiniteEvent v) = 0 := by | |
| intro h | |
| exact (ne_of_gt hp) ((G.percolates_null_iff _).mpr h) | |
| push Not at hn | |
| obtain ⟨v, hv⟩ := hn | |
| have hvpos : 0 < (G.law p).real (G.infiniteEvent v) := | |
| ENNReal.toReal_pos hv (measure_ne_top _ _) | |
| have hall := G.real_infiniteEvent_pos_all hconnected p hp0 hvpos | |
| obtain ⟨S, hS, horbit⟩ := hquasi | |
| have hSne : S.Nonempty := by | |
| obtain ⟨s, hs, _⟩ := horbit v | |
| exact ⟨s, hs⟩ | |
| obtain ⟨s, hs, hmin⟩ := Set.exists_min_image S | |
| (fun o => (G.law p).real (G.infiniteEvent o)) hS hSne | |
| refine ⟨(G.law p).real (G.infiniteEvent s), hall s, ?_⟩ | |
| intro o | |
| obtain ⟨t, ht, φ, rfl⟩ := horbit o | |
| rw [Aut.infiniteEvent_real] | |
| exact hmin t ht | |
| end CriticalPercolation.BondGraph | |
| open Set MeasureTheory ProbabilityTheory Filter | |
| open scoped Topology Classical | |
| end | |
| end | |
| end OAI | |