openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /ProductMeasure /SusceptibilityMeasure.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Clusters.ClusterTail | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.FiniteInterpolationMeasure | |
| import OAI.Probability.CriticalPercolation.Criticality.FiniteSusceptibility | |
| import OAI.Probability.CriticalPercolation.Modular.FiniteClusterTransport | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.FiniteAdjacentMeasure | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set MeasureTheory ProbabilityTheory Filter | |
| open scoped Classical Topology | |
| namespace CriticalPercolation.BondGraph | |
| open FiniteProduct | |
| variable {V E : Type*} (G : BondGraph V E) | |
| lemma finite_clusterTail_measure_bound [Fintype V] [Fintype E] | |
| (p : unitInterval) (D : ℕ) (hD : ∀ v, (G.incidentEdges {v}).ncard≤D) | |
| (hpD : 4*(p:ℝ)*D≤1) (u : V) (s : ℕ) : | |
| (s:ℝ)*(G.law p).real (G.clusterTail u s) ≤ 2 := by | |
| let w := fun _ : E => BernoulliProduct.bitWeight p | |
| have hn : ∀ e b, 0≤w e b := fun _ => BernoulliProduct.bitWeight_nonneg p | |
| have ht := G.finite_susceptibility_bound p p.property w | |
| (fun _ => BernoulliProduct.bitWeight_sum p) hn | |
| (fun _ => by simp only [w,BernoulliProduct.bitWeight,ite_false]) | |
| (fun _ => by simp only [w,BernoulliProduct.bitWeight,ite_true]) D | |
| (fun v => by simpa only [← Set.ncard_eq_toFinset_card'] using hD v) hpD u | |
| have hprob : (G.law p).real (G.clusterTail u s) = prob w (fun ω => ω∈G.clusterTail u s) := by | |
| simpa only [law,← BernoulliProduct.law_const,w] using | |
| (BernoulliProduct.prob_eq_real (fun _ : E => p) (G.clusterTail u s)).symm | |
| rw [hprob,prob,← expect_const_mul] | |
| refine le_trans (expect_mono hn ?_) ht | |
| intro ω | |
| change (s:ℝ)*(if {e | ω e}∈G.clusterTail u s then 1 else 0) ≤ | |
| (G.clusterSize {e | ω e} u : ℝ) | |
| by_cases hh : {e | ω e} ∈ G.clusterTail u s | |
| · simp only [ite_eq_left hh,mul_one] | |
| exact_mod_cast (G.clusterTail_iff_card {e | ω e} u s).mp hh | |
| · simp only [ite_eq_right hh,mul_zero] | |
| exact Nat.cast_nonneg _ | |
| lemma induce_clusterTail_real (S : Set V) (hS : (G.inducedEdges S).Finite) | |
| (p : unitInterval) (u : S) (s : ℕ) : | |
| ((G.induce S).law p).real ((G.induce S).clusterTail u s) = | |
| (G.law p).real {ω | G.inducedEdges S ∩ ω ∈ G.clusterTail u.val s} := by | |
| have he : (G.induce S).clusterTail u s = | |
| {ω | Subtype.val '' ω ∈ G.clusterTail u.val s} := by | |
| ext ω | |
| exact G.induce_clusterTail S ω u s | |
| rw [he,G.induce_law_event S hS] | |
| lemma induced_clusterTail_measure_bound (hlocal : G.LocallyFinite) | |
| (p : unitInterval) (D : ℕ) (hD : ∀ v, (G.incidentEdges {v}).ncard≤D) | |
| (hpD : 4*(p:ℝ)*D≤1) (S : Set V) (hS : S.Finite) (u : S) (s : ℕ) : | |
| (s:ℝ)*(G.law p).real {ω | G.inducedEdges S ∩ ω ∈ G.clusterTail u.val s} ≤ 2 := by | |
| let := hS.fintype | |
| let := (G.finite_inducedEdges hlocal hS).fintype | |
| have h := (G.induce S).finite_clusterTail_measure_bound p D | |
| (G.induce_incident_bound hlocal D hD S) hpD u s | |
| rwa [G.induce_clusterTail_real S (G.finite_inducedEdges hlocal hS)] at h | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end | |
| end OAI | |