| import Mathlib |
| import OAI.Probability.CriticalPercolation.ProductMeasure.LevelCutProbability |
| import OAI.Probability.CriticalPercolation.ProductMeasure.LevelContactMeasurability |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open Set Function MeasureTheory ProbabilityTheory |
| open scoped Classical BigOperators ENNReal |
|
|
| namespace CriticalPercolation.BondGraph |
| open BernoulliProduct |
|
|
| variable {V E : Type*} [Countable V] [Countable E] (G : BondGraph V E) |
|
|
| theorem few_contacts_cost (hlocal : G.LocallyFinite) (q : E → unitInterval) |
| (π : V → ℤ) (j : ℤ) (o : V) (ho : j ≤ π o) |
| (hcross : ∀ e v w, G.ends e=s(v,w) → j ≤ π v → π w<j → π v=j) |
| (p : unitInterval) |
| (hq : ∀ e v w, G.ends e=s(v,w) → π v=j → π w<j → (q e : ℝ) ≤ p) |
| (D k : ℕ) (hD : ∀ v, π v=j → (G.incidentEdges {v}).ncard ≤ D) : |
| (1-(p : ℝ))^(D*k)*(BernoulliProduct.law q).real (G.fewContacts π j o k) ≤ |
| (BernoulliProduct.law q).real (G.minimumLevel π o j) := by |
| let X := G.inducedEdges {v | j ≤ π v} |
| let d : Set E → Finset E := fun ω => finiteSetChoice (G.contactDrop π j o ω) |
| let A : Finset E → Set (Set E) := fun U => closedBlock ((U : Set E)∩Xᶜ) |
| have hd (U : Finset E) : MeasurableSet[blockSigma X] {ω | d ω=U} := |
| @measurableSet_finiteSetChoice (Set E) E (blockSigma X) inferInstance |
| (G.contactDrop π j o) (G.measurable_contactDrop π j o) U |
| have hA (U : Finset E) : MeasurableSet[blockSigma Xᶜ] (A U) := |
| (blockSigma_mono Set.inter_subset_right) _ |
| (measurableSet_closedBlock _ ((U.finite_toSet.subset Set.inter_subset_left).countable)) |
| have he (ω : Set E) (hω : ω∈G.fewContacts π j o k) : |
| ((d ω : Set E)∩Xᶜ)=G.contactDrop π j o ω := by |
| rw [finiteSetChoice_coe (G.finite_contactDrop hlocal π j o ω |
| (G.fewContacts_finite π j o k ω hω).1)] |
| exact Set.inter_eq_left.mpr (Set.disjoint_left.mp |
| (G.contactDrop_disjoint_exterior π j o ω)) |
| have hb (ω : Set E) (hω : ω∈G.fewContacts π j o k) : |
| (1-(p : ℝ))^(D*k) ≤ (BernoulliProduct.law q).real (A (d ω)) := by |
| change _ ≤ (BernoulliProduct.law q).real (closedBlock ((d ω : Set E)∩Xᶜ)) |
| rw [he ω hω] |
| have hω' := G.fewContacts_finite π j o k ω hω |
| apply law_closedBlock_lower q _ (G.finite_contactDrop hlocal π j o ω hω'.1) p |
| · calc |
| _ ≤ (G.levelContacts π j o ω).ncard*D := |
| G.contactDrop_ncard_le hlocal π j o ω hω'.1 D (fun v hv => hD v hv.2) |
| _ ≤ k*D := Nat.mul_le_mul_right D (Nat.le_of_lt hω'.2.2) |
| _ = D*k := Nat.mul_comm _ _ |
| · rintro e ⟨v,hv,w,hw,hend⟩ |
| exact hq e v w hend hv.2 hw |
| have hh := selected_fresh_lower q X Xᶜ disjoint_compl_right d A hd hA |
| (G.fewContacts π j o k) (G.measurableSet_fewContacts π j o k) |
| ((1-(p : ℝ))^(D*k)) (pow_nonneg (sub_nonneg.mpr p.2.2) _) hb |
| apply hh.trans |
| refine measureReal_mono ?_ (measure_ne_top _ _) |
| rintro ω ⟨hω,hclosed⟩ |
| apply G.minimumLevel_of_drop_closed π j o ho hcross ω |
| (G.fewContacts_finite π j o k ω hω).2.1 |
| change Disjoint ((d ω : Set E)∩Xᶜ) ω at hclosed |
| rwa [he ω hω] at hclosed |
|
|
| theorem exists_few_contact_level (hlocal : G.LocallyFinite) (q : E → unitInterval) |
| (π : V → ℤ) (o : V) (J : Finset ℤ) (hJ : J.Nonempty) |
| (ho : ∀ j∈J, j ≤ π o) |
| (hcross : ∀ j∈J, ∀ e v w, G.ends e=s(v,w) → j ≤ π v → π w<j → π v=j) |
| (p : unitInterval) (hp : (p : ℝ)<1) |
| (hq : ∀ j∈J, ∀ e v w, G.ends e=s(v,w) → π v=j → π w<j → (q e : ℝ) ≤ p) |
| (D k : ℕ) (hD : ∀ j∈J, ∀ v, π v=j → (G.incidentEdges {v}).ncard ≤ D) : |
| ∃ j∈J, (BernoulliProduct.law q).real (G.fewContacts π j o k) ≤ |
| 1/((J.card : ℝ)*(1-(p : ℝ))^(D*k)) := by |
| apply exists_level_small (BernoulliProduct.law q) J hJ (fun j => G.fewContacts π j o k) |
| (G.minimumLevel π o) (fun j _ => G.measurableSet_minimumLevel π o j) |
| · intro i hi j hj hij |
| exact G.minimumLevel_pairwiseDisjoint π o hij |
| · exact pow_pos (sub_pos.mpr hp) _ |
| · intro j hj |
| exact G.few_contacts_cost hlocal q π j o (ho j hj) (hcross j hj) p (hq j hj) D k (hD j hj) |
|
|
| end CriticalPercolation.BondGraph |
| end |
| end |
| end |
|
|
| end OAI |
|
|
