| import Mathlib |
| import OAI.Probability.CriticalPercolation.ProductMeasure.MonotoneIndicatorUpper |
| import OAI.Probability.CriticalPercolation.Modular.ModularClusterMass |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set MeasureTheory Filter |
| open scoped Classical ENNReal Topology |
|
|
| namespace CriticalPercolation.BondGraph |
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| def massTailEvent (o v : V) (a : ℝ≥0∞) : Set (Set E) := |
| {ω | a<G.modularMass o (G.cluster ω v)} |
|
|
| lemma massTailEvent_upper (o v : V) (a : ℝ≥0∞) : IsUpperSet (G.massTailEvent o v a) := |
| fun _ _ h hh => hh.trans_le (G.modularMass_mono o (G.cluster_mono h v)) |
|
|
| lemma massTailEvent_finiteWitness (o v : V) (a : ℝ≥0∞) : |
| BernoulliProduct.HasFiniteOpenWitness (G.massTailEvent o v a) := by |
| intro ω hω |
| change a<∑' x, if x∈G.cluster ω v then G.stabilizerWeight o x else 0 at hω |
| rw [ENNReal.tsum_eq_iSup_sum,lt_iSup_iff] at hω |
| obtain ⟨S,hS⟩ := hω |
| let T : Set V := (S : Set V) ∩ G.cluster ω v |
| obtain ⟨s,hs,hsω,hTs⟩ := G.finite_subset_cluster_finite_witness |
| (S.finite_toSet.inter_of_left _) (show T⊆G.cluster ω v from Set.inter_subset_right) |
| refine ⟨s,hs,hsω,lt_of_lt_of_le hS ?_⟩ |
| change (∑ x∈S, if x∈G.cluster ω v then G.stabilizerWeight o x else 0) ≤ |
| ∑' x, if x∈G.cluster s v then G.stabilizerWeight o x else 0 |
| apply (Finset.sum_le_sum (fun x hx => ?_)).trans (ENNReal.sum_le_tsum S) |
| by_cases hxω : x∈G.cluster ω v |
| · rw [ite_eq_left hxω,ite_eq_left (hTs ⟨hx,hxω⟩)] |
| · simp only [ite_eq_right hxω,zero_le] |
|
|
| lemma measurableSet_massTailEvent [Countable V] [Countable E] (o v : V) (a : ℝ≥0∞) : |
| MeasurableSet (G.massTailEvent o v a) := |
| measurableSet_lt measurable_const (G.measurable_modularClusterMass o v) |
|
|
| lemma heavyClusterEvent_eq_iInter_massTail (o v : V) : |
| G.heavyClusterEvent o v=⋂ n : ℕ, G.massTailEvent o v n := by |
| ext ω |
| simp only [heavyClusterEvent,massTailEvent,Set.mem_ofPred_eq,Set.mem_iInter] |
| constructor |
| · intro h n |
| rw [h] |
| exact ENNReal.natCast_lt_top n |
| · intro h |
| by_contra hn |
| obtain ⟨n,hnn⟩ := exists_nat_gt (G.modularMass o (G.cluster ω v)).toReal |
| have hlt : (n:ℝ≥0∞)<G.modularMass o (G.cluster ω v) := h n |
| have hh := (ENNReal.toReal_lt_toReal (ENNReal.natCast_ne_top n) hn).mpr hlt |
| simp only [ENNReal.toReal_natCast] at hh |
| exact (lt_irrefl _) (hh.trans hnn) |
|
|
| lemma massTailEvent_antitone (o v : V) : Antitone (fun n : ℕ => G.massTailEvent o v n) := by |
| intro n m h ω hω |
| exact lt_of_le_of_lt (show (n : ℝ≥0∞) ≤ (m : ℝ≥0∞) by exact_mod_cast h) hω |
|
|
| lemma tendsto_real_massTail [Countable V] [Countable E] (p : unitInterval) (o v : V) : |
| Tendsto (fun n : ℕ => (G.law p).real (G.massTailEvent o v n)) atTop |
| (𝓝 ((G.law p).real (G.heavyClusterEvent o v))) := by |
| have h := tendsto_measure_iInter_atTop (μ := G.law p) |
| (fun n : ℕ => (G.measurableSet_massTailEvent o v n).nullMeasurableSet) |
| (G.massTailEvent_antitone o v) ⟨0,measure_ne_top _ _⟩ |
| rw [←G.heavyClusterEvent_eq_iInter_massTail] at h |
| exact (ENNReal.tendsto_toReal (measure_ne_top _ _)).comp h |
|
|
| lemma harris_heavyClusterEvent [Countable V] [Countable E] (p : unitInterval) (o x y : V) : |
| (G.law p).real (G.heavyClusterEvent o x) * (G.law p).real (G.heavyClusterEvent o y) ≤ |
| (G.law p).real (G.heavyClusterEvent o x ∩ G.heavyClusterEvent o y) := by |
| have hlim := tendsto_measure_iInter_atTop (μ := G.law p) |
| (fun n : ℕ => ((G.measurableSet_massTailEvent o x n).inter |
| (G.measurableSet_massTailEvent o y n)).nullMeasurableSet) |
| (show Antitone (fun n : ℕ => G.massTailEvent o x n ∩ G.massTailEvent o y n) from |
| fun _ _ h => Set.inter_subset_inter (G.massTailEvent_antitone o x h) |
| (G.massTailEvent_antitone o y h)) ⟨0,measure_ne_top _ _⟩ |
| rw [Set.iInter_inter_distrib,←G.heavyClusterEvent_eq_iInter_massTail, |
| ←G.heavyClusterEvent_eq_iInter_massTail] at hlim |
| apply le_of_tendsto_of_tendsto |
| ((G.tendsto_real_massTail p o x).mul (G.tendsto_real_massTail p o y)) |
| ((ENNReal.tendsto_toReal (measure_ne_top _ _)).comp hlim) |
| apply Eventually.of_forall |
| intro n |
| simpa only [BernoulliProduct.law_const,BondGraph.law,measureReal_def,Function.comp_apply] using |
| BernoulliProduct.harris_finiteWitness (fun _ : E => p) |
| (G.massTailEvent o x n) (G.massTailEvent o y n) |
| (G.massTailEvent_upper o x n) (G.massTailEvent_upper o y n) |
| (G.massTailEvent_finiteWitness o x n) (G.massTailEvent_finiteWitness o y n) |
|
|
| end CriticalPercolation.BondGraph |
| end |
| end |
|
|
| end OAI |
|
|
