| import Mathlib |
| import OAI.Probability.CriticalPercolation.GraphGeometry.OpenGraphIso |
| import OAI.Probability.CriticalPercolation.ProductMeasure.MeasurableSetRestrict |
|
|
| namespace OAI |
|
|
| section |
| section |
| open scoped BigOperators |
| open scoped Classical ENNReal |
| open Set |
| open Set MeasureTheory ProbabilityTheory |
| open scoped Classical |
| open Filter Topology |
| open scoped Topology Classical |
| open Set MeasureTheory ProbabilityTheory Filter |
| open scoped Classical Topology |
|
|
| namespace CriticalPercolation.BondGraph |
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| def clusterTail (o : V) (n : ℕ) : Set (Set E) := |
| {ω | ∃ K : Set V, K.Finite ∧ K ⊆ G.cluster ω o ∧ n ≤ K.ncard} |
|
|
| lemma clusterTail_isUpperSet (o : V) (n : ℕ) : IsUpperSet (G.clusterTail o n) := by |
| rintro ω ω' h ⟨K, hK, hKω, hn⟩ |
| exact ⟨K, hK, hKω.trans (G.cluster_mono h o), hn⟩ |
|
|
| lemma clusterTail_hasFiniteOpenWitness (o : V) (n : ℕ) : |
| BernoulliProduct.HasFiniteOpenWitness (G.clusterTail o n) := by |
| rintro ω ⟨K, hK, hKω, hn⟩ |
| obtain ⟨s, hs, hsω, hKs⟩ := G.finite_subset_cluster_finite_witness hK hKω |
| exact ⟨s, hs, hsω, K, hK, hKs, hn⟩ |
|
|
| lemma measurableSet_clusterTail [Countable E] (o : V) (n : ℕ) : |
| MeasurableSet (G.clusterTail o n) := |
| BernoulliProduct.measurableSet_finiteWitness _ (G.clusterTail_isUpperSet o n) |
| (G.clusterTail_hasFiniteOpenWitness o n) |
|
|
| lemma clusterTail_antitone (o : V) : Antitone (G.clusterTail o) := by |
| rintro m n h ω ⟨K, hK, hKω, hn⟩ |
| exact ⟨K, hK, hKω, h.trans hn⟩ |
|
|
| lemma infiniteEvent_subset_clusterTail (o : V) (n : ℕ) : |
| G.infiniteEvent o ⊆ G.clusterTail o n := by |
| intro ω hω |
| obtain ⟨K, hKω, hK, hn⟩ := hω.exists_subset_ncard_eq n |
| exact ⟨K, hK, hKω, hn.ge⟩ |
|
|
| lemma infiniteEvent_eq_iInter_clusterTail (o : V) : |
| G.infiniteEvent o = ⋂ n, G.clusterTail o n := by |
| apply Set.Subset.antisymm |
| · exact Set.subset_iInter fun n => G.infiniteEvent_subset_clusterTail o n |
| · intro ω hω |
| change (G.cluster ω o).Infinite |
| intro hfin |
| obtain ⟨K, hK, hKω, hn⟩ := Set.mem_iInter.mp hω ((G.cluster ω o).ncard + 1) |
| have hle := Set.ncard_le_ncard hKω hfin |
| omega |
|
|
| lemma tendsto_real_clusterTail [Countable E] (p : unitInterval) (o : V) : |
| Tendsto (fun n => (G.law p).real (G.clusterTail o n)) atTop |
| (𝓝 ((G.law p).real (G.infiniteEvent o))) := by |
| have h := tendsto_measure_iInter_atTop (μ := G.law p) |
| (fun n => (G.measurableSet_clusterTail o n).nullMeasurableSet) |
| (G.clusterTail_antitone o) |
| ⟨0, measure_ne_top _ _⟩ |
| rw [← G.infiniteEvent_eq_iInter_clusterTail o] at h |
| exact (ENNReal.tendsto_toReal (measure_ne_top _ _)).comp h |
|
|
| lemma lowerSemicontinuous_real_clusterTail [Countable E] (o : V) (n : ℕ) : |
| LowerSemicontinuous (fun p : unitInterval => (G.law p).real (G.clusterTail o n)) := by |
| simpa only [law, BernoulliProduct.law_const] using |
| BernoulliProduct.lowerSemicontinuous_real_finiteWitness _ |
| (G.clusterTail_isUpperSet o n) (G.clusterTail_hasFiniteOpenWitness o n) |
|
|
| lemma Aut.preimage_clusterTail (φ : G.Aut) (o : V) (n : ℕ) : |
| (fun ω : Set E => φ.edge '' ω) ⁻¹' G.clusterTail (φ.vertex o) n = |
| G.clusterTail o n := by |
| ext ω |
| simp only [Set.mem_preimage, clusterTail, Set.mem_ofPred_eq] |
| rw [← φ.cluster_image] |
| constructor |
| · rintro ⟨K, hK, hKω, hn⟩ |
| refine ⟨φ.vertex.symm '' K, hK.image _, ?_, ?_⟩ |
| · rintro _ ⟨v, hv, rfl⟩ |
| obtain ⟨w, hw, rfl⟩ := hKω hv |
| simpa using hw |
| · rwa [Set.ncard_image_of_injective _ φ.vertex.symm.injective] |
| · rintro ⟨K, hK, hKω, hn⟩ |
| exact ⟨φ.vertex '' K, hK.image _, Set.image_mono hKω, |
| by rwa [Set.ncard_image_of_injective _ φ.vertex.injective]⟩ |
|
|
| lemma Aut.clusterTail_real [Countable E] (φ : G.Aut) (p : unitInterval) (o : V) (n : ℕ) : |
| (G.law p).real (G.clusterTail (φ.vertex o) n) = (G.law p).real (G.clusterTail o n) := by |
| simp only [measureReal_def] |
| congr 1 |
| conv_lhs => rw [← G.law_map_aut p φ] |
| rw [Measure.map_apply (BernoulliProduct.measurable_image_equiv φ.edge) |
| (G.measurableSet_clusterTail _ _), φ.preimage_clusterTail] |
|
|
| theorem subcritical_uniform_clusterTail |
| (hconnected : G.fullGraph.Connected) (hlocal : G.LocallyFinite) |
| (hquasi : G.QuasiTransitive) (p : unitInterval) (hp : 0 < p) |
| (θ : ℝ) (hθ : 0 < θ) |
| (hroot : ∀ o, θ ≤ (G.law p).real (G.infiniteEvent o)) (L : ℕ) : |
| ∃ q : unitInterval, (p : ℝ) / 2 ≤ q ∧ q < p ∧ |
| ∀ o, 3 * θ / 4 < (G.law q).real (G.clusterTail o L) := by |
| let := G.countable_vertices hconnected hlocal |
| let := G.countable_bonds hconnected hlocal |
| obtain ⟨S, hS, hcover⟩ := hquasi |
| let := hS.fintype |
| obtain ⟨q, hhalf, hqp, hq⟩ := BernoulliProduct.exists_lower_parameter |
| (fun o : S => G.clusterTail o.val L) |
| (fun o => G.clusterTail_isUpperSet o.val L) |
| (fun o => G.clusterTail_hasFiniteOpenWitness o.val L) |
| p hp (fun _ => 3 * θ / 4) (by |
| intro o |
| rw [BernoulliProduct.law_const] |
| change 3 * θ / 4 < (G.law p).real (G.clusterTail o.val L) |
| have hn := measureReal_mono (μ := G.law p) (G.infiniteEvent_subset_clusterTail o.val L) |
| have ho := hroot o.val |
| linarith) |
| refine ⟨q, hhalf, hqp, fun o => ?_⟩ |
| obtain ⟨v, hv, φ, rfl⟩ := hcover o |
| rw [φ.clusterTail_real] |
| simpa only [BernoulliProduct.law_const, law] using hq ⟨v, hv⟩ |
|
|
| lemma tendsto_real_clusterTail_subcritical [Countable V] [Countable E] |
| {p : unitInterval} (hp : p < G.criticalProbability) (o : V) : |
| Tendsto (fun n => (G.law p).real (G.clusterTail o n)) atTop (𝓝 0) := by |
| have hz : G.law p (G.infiniteEvent o) = 0 := by |
| exact measure_mono_null (show G.infiniteEvent o ⊆ G.percolates from |
| fun ω hω => ⟨o, hω⟩) (G.law_percolates_eq_zero_of_lt_critical hp) |
| simpa only [measureReal_def, hz, ENNReal.toReal_zero] using G.tendsto_real_clusterTail p o |
|
|
| end CriticalPercolation.BondGraph |
| open Set MeasureTheory ProbabilityTheory Filter |
| open scoped Classical Topology |
|
|
| end |
| end |
|
|
| end OAI |
|
|
