openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Modular /ClusterMassEndpoint.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Clusters.ClusterTail | |
| import OAI.Probability.CriticalPercolation.Criticality.ExhaustionEvents | |
| import OAI.Probability.CriticalPercolation.Modular.ClusterMassRestriction | |
| import OAI.Probability.CriticalPercolation.Modular.ClusterMassEndpointFinite | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set MeasureTheory Filter | |
| open scoped Classical BigOperators Topology | |
| namespace CriticalPercolation.FiniteClusters | |
| variable {V E : Type*} (G : BondGraph V E) | |
| lemma restrict_edgeDiscrepancy_eq {K ω : Set E} (e : K) | |
| (hx : G.incidentEdges (G.cluster ω (G.ends e.val).out.1) ⊆ K) | |
| (hy : G.incidentEdges (G.cluster ω (G.ends e.val).out.2) ⊆ K) | |
| (s : ℕ) (p : ℝ) : | |
| edgeDiscrepancy (G.restrictEdges K) s p (Subtype.val ⁻¹' ω) e = | |
| edgeDiscrepancy G s p ω e.val := by | |
| have hc := cluster_inter_eq G hx | |
| have hconn : ((G.restrictEdges K).openGraph (Subtype.val ⁻¹' ω)).Reachable | |
| (G.ends e.val).out.1 (G.ends e.val).out.2 ↔ | |
| (G.openGraph ω).Reachable (G.ends e.val).out.1 (G.ends e.val).out.2 := by | |
| rw [G.restrictEdges_openGraph] | |
| exact Set.ext_iff.mp hc _ | |
| unfold edgeDiscrepancy edgeMass size coefficient | |
| change (if e.val ∈ ω then -(1-p)/p else 1) * | |
| (cappedMass s ((G.restrictEdges K).cluster (Subtype.val ⁻¹' ω) (G.ends e.val).out.1).ncard + | |
| (if ((G.restrictEdges K).openGraph (Subtype.val ⁻¹' ω)).Reachable (G.ends e.val).out.1 (G.ends e.val).out.2 then 0 | |
| else cappedMass s ((G.restrictEdges K).cluster (Subtype.val ⁻¹' ω) (G.ends e.val).out.2).ncard)) = _ | |
| rw [hconn,restrict_cluster,restrict_cluster,cluster_inter_eq G hx,cluster_inter_eq G hy] | |
| lemma restrict_twoCluster (K : Set E) (ω : Set E) (u v : V) (s : ℕ) : | |
| Subtype.val ⁻¹' ω ∈ (G.restrictEdges K).twoClusterEvent u v s ↔ | |
| K ∩ ω ∈ G.twoClusterEvent u v s := by | |
| simp only [BondGraph.twoClusterEvent,Set.mem_inter_iff,BondGraph.clusterTail, | |
| Set.mem_ofPred_eq,restrict_cluster,G.restrictEdges_openGraph] | |
| variable [Countable V] [Countable E] | |
| lemma restrict_twoCluster_real (p : unitInterval) (K : Set E) (u v : V) (s : ℕ) : | |
| ((G.restrictEdges K).law p).real ((G.restrictEdges K).twoClusterEvent u v s) = | |
| (G.law p).real {ω | K ∩ ω ∈ G.twoClusterEvent u v s} := by | |
| simp only [BondGraph.law,← BernoulliProduct.law_const] | |
| rw [← BernoulliProduct.law_map_restrict (fun _ : E => p) K] | |
| simp only [measureReal_def] | |
| rw [Measure.map_apply (measurable_restrict K) | |
| ((G.restrictEdges K).measurableSet_twoClusterEvent u v s)] | |
| congr 2 | |
| ext ω | |
| exact restrict_twoCluster G K ω u v s | |
| lemma restricted_edgeDiscrepancy_integral (p : unitInterval) (K : Set E) (e : K) (s : ℕ) : | |
| (∫ ω, edgeDiscrepancy (G.restrictEdges K) s p (Subtype.val ⁻¹' ω) e ∂G.law p) = | |
| ∫ ω, edgeDiscrepancy (G.restrictEdges K) s p ω e ∂(G.restrictEdges K).law p := by | |
| simp only [BondGraph.law,← BernoulliProduct.law_const] | |
| rw [← BernoulliProduct.law_map_restrict (fun _ : E => p) K] | |
| rw [integral_map (measurable_restrict K).aemeasurable | |
| (measurable_edgeDiscrepancy (G.restrictEdges K) s p e).aestronglyMeasurable] | |
| theorem subcritical_endpoint_le_integral (hl : G.LocallyFinite) | |
| (s : ℕ) (hs : 0<s) (p : unitInterval) (hp : (p:ℝ) ∈ Ioo 0 1) | |
| (hpc : p < G.criticalProbability) (e : E) : | |
| (G.law p).real (G.twoClusterEvent (G.ends e).out.1 (G.ends e).out.2 s) ≤ | |
| ∫ ω, edgeDiscrepancy G s p ω e ∂G.law p := by | |
| let K₀ := (Set.to_countable (Set.univ : Set E)).finiteExhaustion | |
| let K : Set.FiniteExhaustion (Set.univ : Set E) := | |
| { toFun := fun n => insert e (K₀ n) | |
| finite' := fun n => (K₀.finite n).insert e | |
| subset_succ' := fun n => Set.insert_subset_insert (K₀.subset_succ n) | |
| iUnion_eq' := by | |
| apply Set.eq_univ_of_forall | |
| intro a | |
| obtain ⟨n,hn⟩ := Set.mem_iUnion.mp (K₀.iUnion_eq.symm ▸ Set.mem_univ a) | |
| exact Set.mem_iUnion.mpr ⟨n,Or.inr hn⟩ } | |
| let en (n : ℕ) : K n := ⟨e,Or.inl rfl⟩ | |
| let F := fun n ω => edgeDiscrepancy (G.restrictEdges (K n)) s p (Subtype.val ⁻¹' ω) (en n) | |
| have hb (n : ℕ) : (G.law p).real {ω | K n ∩ ω ∈ | |
| G.twoClusterEvent (G.ends e).out.1 (G.ends e).out.2 s} ≤ ∫ ω, F n ω ∂G.law p := by | |
| rw [← restrict_twoCluster_real G p (K n),show (∫ ω, F n ω ∂G.law p) = _ from | |
| restricted_edgeDiscrepancy_integral G p (K n) (en n) s] | |
| let := (K.finite n).fintype | |
| exact finite_endpoint_le_integral (G.restrictEdges (K n)) s hs p hp (en n) | |
| have ht : Tendsto (fun n => ∫ ω, F n ω ∂G.law p) atTop | |
| (𝓝 (∫ ω, edgeDiscrepancy G s p ω e ∂G.law p)) := by | |
| apply tendsto_integral_of_dominated_convergence (fun _ => 2/(p:ℝ)) | |
| · intro n | |
| exact ((measurable_edgeDiscrepancy (G.restrictEdges (K n)) s p (en n)).comp | |
| (measurable_restrict (K n))).aestronglyMeasurable | |
| · exact integrable_const _ | |
| · intro n | |
| apply Eventually.of_forall | |
| intro ω | |
| exact edgeDiscrepancy_abs_le (G.restrictEdges (K n)) s hp _ (en n) | |
| · filter_upwards [ae_finite_clusters G p hpc] with ω hω | |
| apply tendsto_const_nhds.congr' | |
| filter_upwards [eventually_incident_subset G hl K (hω (G.ends e).out.1), | |
| eventually_incident_subset G hl K (hω (G.ends e).out.2)] with n hn hm | |
| exact (restrict_edgeDiscrepancy_eq G (en n) hn hm s p).symm | |
| exact le_of_tendsto_of_tendsto (G.tendsto_real_twoCluster_restrict p _ _ s K) ht | |
| (Eventually.of_forall hb) | |
| end CriticalPercolation.FiniteClusters | |
| end | |
| end | |
| end OAI | |