openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Clusters /ForcedDisconnectContinuity.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.MonotoneBernoulli | |
| import OAI.Probability.CriticalPercolation.Clusters.ClusterCountFlip | |
| import OAI.Probability.CriticalPercolation.Modular.PeakReturn | |
| import OAI.Probability.CriticalPercolation.Modular.CriticalHeavyIdentity | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set MeasureTheory Filter | |
| open scoped Classical ENNReal Topology BigOperators | |
| namespace CriticalPercolation.BondGraph | |
| open BernoulliProduct | |
| variable {V E : Type*} (G : BondGraph V E) | |
| def forcedClosedSeparated (ω : Set E) (e : E) : Prop := | |
| ∃ a∈G.ends e, ∃ b∈G.ends e, ¬(G.openGraph (ω\{e})).Reachable a b | |
| def disconnectedIndicator (ω : Set E) (e : E) : ℝ := | |
| if G.forcedClosedSeparated ω e then 1 else 0 | |
| lemma forcedClosedSeparated_of_ends (ω : Set E) (e : E) (a b : V) (he : G.ends e=s(a,b)) : | |
| G.forcedClosedSeparated ω e ↔ ¬(G.openGraph (ω\{e})).Reachable a b := by | |
| unfold forcedClosedSeparated | |
| constructor | |
| · rintro ⟨x,hx,y,hy,hxy⟩ hab | |
| rw [he,Sym2.mem_iff] at hx hy | |
| rcases hx with rfl|rfl <;> rcases hy with rfl|rfl | |
| · exact hxy (.refl _) | |
| · exact hxy hab | |
| · exact hxy hab.symm | |
| · exact hxy (.refl _) | |
| · intro h | |
| exact ⟨a,by simp [he],b,by simp [he],h⟩ | |
| lemma disconnectedIndicator_of_ends (ω : Set E) (e : E) (a b : V) (he : G.ends e=s(a,b)) : | |
| G.disconnectedIndicator ω e=if ¬(G.openGraph (ω\{e})).Reachable a b then 1 else 0 := by | |
| simp only [disconnectedIndicator,G.forcedClosedSeparated_of_ends ω e a b he] | |
| lemma measurable_forcedClosedSeparated [Countable V] [Countable E] (e : E) : | |
| Measurable (fun ω => G.forcedClosedSeparated ω e) := by | |
| unfold forcedClosedSeparated | |
| apply Measurable.exists | |
| intro a | |
| apply measurable_const.and | |
| apply Measurable.exists | |
| intro b | |
| apply measurable_const.and | |
| exact ((G.measurable_reachable a b).comp (by | |
| have h := measurable_forceOn ({e}:Set E) ∅ | |
| change Measurable (fun ω => forceOn {e} ∅ ω) at h | |
| simpa only [forceOn_empty_singleton] using h)).not | |
| lemma measurable_disconnectedIndicator [Countable V] [Countable E] (e : E) : | |
| Measurable (fun ω => G.disconnectedIndicator ω e) := | |
| measurable_const.ite (G.measurable_forcedClosedSeparated e).setOf measurable_const | |
| lemma norm_disconnectedIndicator_le (ω : Set E) (e : E) : ‖G.disconnectedIndicator ω e‖≤1 := by | |
| unfold disconnectedIndicator | |
| split_ifs <;> norm_num | |
| lemma clusterCountDrop_eq_disconnected_of_finite (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) | |
| (ω : Set E) (e : E) (hω : ∀ x, (G.cluster ω x).Finite) : | |
| G.clusterCountDrop ω e=G.disconnectedIndicator ω e := by | |
| obtain ⟨⟨a,b⟩,he⟩ := Quot.exists_rep (G.ends e) | |
| rw [G.clusterCountDrop_of_ends hc hl ω e a b he.symm, | |
| G.disconnectedIndicator_of_ends ω e a b he.symm] | |
| have ha : G.modularMass a (G.cluster (ω\{e}) a)≠∞ := | |
| (G.modularMass_finite a ((hω a).subset (G.cluster_mono Set.sdiff_subset a))).ne | |
| simp only [ha,false_and,not_false_eq_true,and_true] | |
| lemma clusterCountDrop_le_disconnected (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) | |
| (ω : Set E) (e : E) : G.clusterCountDrop ω e≤G.disconnectedIndicator ω e := by | |
| obtain ⟨⟨a,b⟩,he⟩ := Quot.exists_rep (G.ends e) | |
| rw [G.clusterCountDrop_of_ends hc hl ω e a b he.symm, | |
| G.disconnectedIndicator_of_ends ω e a b he.symm] | |
| split_ifs <;> norm_num at * | |
| tauto | |
| lemma finite_clusters_ae_of_subcritical [Countable V] [Countable E] (p : unitInterval) | |
| (hp : p<G.criticalProbability) : ∀ᵐ ω ∂G.law p, ∀ x, (G.cluster ω x).Finite := by | |
| have h := G.law_percolates_eq_zero_of_lt_critical hp | |
| have ha : ∀ᵐ ω ∂G.law p, ω∉G.percolates := by | |
| rw [ae_iff] | |
| simpa only [not_not,Set.ofPred_mem_eq] using h | |
| filter_upwards [ha] with ω hω x | |
| by_contra hf | |
| exact hω ⟨x,hf⟩ | |
| lemma clusterCountDrop_eq_disconnected_ae_subcritical [Countable V] [Countable E] | |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (p : unitInterval) | |
| (hp : p<G.criticalProbability) (e : E) : | |
| (fun ω => G.clusterCountDrop ω e)=ᵐ[G.law p] (fun ω => G.disconnectedIndicator ω e) := by | |
| filter_upwards [G.finite_clusters_ae_of_subcritical p hp] with ω hω | |
| exact G.clusterCountDrop_eq_disconnected_of_finite hc hl ω e hω | |
| lemma eventually_disconnectedIndicator (a : ℕ → Set E) (ω : Set E) | |
| (ha : Monotone a) (hu : ⋃ n, a n=ω) (e : E) : | |
| ∀ᶠ n in atTop, G.disconnectedIndicator (a n) e=G.disconnectedIndicator ω e := by | |
| have hmono : Monotone (fun n => a n\{e}) := fun _ _ h => Set.sdiff_subset_sdiff_left (ha h) | |
| have hU : (⋃ n, a n\{e})=ω\{e} := by | |
| rw [←Set.iUnion_sdiff,hu] | |
| obtain ⟨⟨x,y⟩,he⟩ := Quot.exists_rep (G.ends e) | |
| simp_rw [G.disconnectedIndicator_of_ends _ e x y he.symm] | |
| by_cases h : (G.openGraph (ω\{e})).Reachable x y | |
| · filter_upwards [G.eventually_cluster_of_exhaustion (fun n => a n\{e}) (ω\{e}) hmono hU h] with n hn | |
| change (G.openGraph (a n\{e})).Reachable x y at hn | |
| simp only [hn,h,not_true_eq_false,ite_false] | |
| · apply Eventually.of_forall | |
| intro n | |
| have hn : ¬(G.openGraph (a n\{e})).Reachable x y := fun hn => h | |
| (G.cluster_mono (Set.sdiff_subset_sdiff_left (hu ▸ Set.subset_iUnion a n)) x hn) | |
| simp only [hn,h,not_false_eq_true,ite_true] | |
| lemma integral_atParameter [Countable V] [Countable E] (p : unitInterval) | |
| (f : Set E → ℝ) (hf : Measurable f) : | |
| (∫ u, f (atParameter p u) ∂markLaw) = ∫ ω, f ω ∂G.law p := by | |
| rw [←integral_map (measurable_atParameter p).aemeasurable hf.aestronglyMeasurable, | |
| markLaw_map_atParameter,BernoulliProduct.law_const] | |
| rfl | |
| theorem tendsto_integral_disconnectedIndicator [Countable V] [Countable E] | |
| (p : unitInterval) (q : ℕ → unitInterval) (hmono : Monotone q) | |
| (hqp : ∀ n, q n≤p) (hlim : Tendsto q atTop (𝓝 p)) (e : E) : | |
| Tendsto (fun n => ∫ ω, G.disconnectedIndicator ω e ∂G.law (q n)) atTop | |
| (𝓝 (∫ ω, G.disconnectedIndicator ω e ∂G.law p)) := by | |
| have hm (r : unitInterval) : Measurable (fun u => G.disconnectedIndicator (atParameter r u) e) := | |
| (G.measurable_disconnectedIndicator e).comp (measurable_atParameter r) | |
| have ht (u : E → ℝ) : Tendsto (fun n => G.disconnectedIndicator (atParameter (q n) u) e) atTop | |
| (𝓝 (G.disconnectedIndicator (atParameter p u) e)) := | |
| tendsto_const_nhds.congr' (Filter.EventuallyEq.symm (G.eventually_disconnectedIndicator (fun n => atParameter (q n) u) | |
| (atParameter p u) ((atParameter_mono u).comp hmono) (atParameter_iUnion p q hqp hlim u) e)) | |
| have h := tendsto_integral_of_dominated_convergence (μ := (markLaw : Measure (E → ℝ))) | |
| (fun _ => (1:ℝ)) (fun n => (hm (q n)).aestronglyMeasurable) (integrable_const 1) | |
| (fun n => Eventually.of_forall (fun u => G.norm_disconnectedIndicator_le _ e)) (Eventually.of_forall ht) | |
| simpa only [G.integral_atParameter _ _ (G.measurable_disconnectedIndicator e)] using h | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end OAI | |