| import Mathlib |
| import OAI.Probability.CriticalPercolation.Model |
| import OAI.Probability.CriticalPercolation.ProductMeasure.TailProduct |
| import OAI.Probability.CriticalPercolation.ProductMeasure.LevelContactMeasurability |
| import OAI.Probability.CriticalPercolation.Gluing.HonestChoice |
| import OAI.Probability.CriticalPercolation.Gluing.TrialSelection |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open Set Function MeasureTheory |
| open scoped Classical BigOperators |
|
|
| namespace CriticalPercolation |
|
|
| lemma measurableSet_finset_subset {Ω V : Type*} {m : MeasurableSpace Ω} |
| (S : Ω → Set V) (hS : @Measurable Ω (Set V) m inferInstance S) (U : Finset V) : |
| MeasurableSet[m] {ω | (U : Set V)⊆S ω} := by |
| have he : {ω | (U : Set V)⊆S ω}=⋂ v∈U, {ω | v∈S ω} := by |
| ext ω |
| simp [Set.subset_def] |
| rw [he] |
| exact MeasurableSet.biInter U.countable_toSet (fun v _ => |
| ((measurable_pi_apply v).comp hS).setOf) |
|
|
| namespace BondGraph |
| open BernoulliProduct |
|
|
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| lemma levelContacts_reachable (π : V → ℤ) (j : ℤ) (o : V) (ω : Set E) |
| {v : V} (hv : v∈G.levelContacts π j o ω) : (G.openGraph ω).Reachable o v := |
| G.cluster_mono Set.inter_subset_left o hv.1 |
|
|
| lemma levelContacts_nonempty_of_hit (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) |
| (ω : Set E) (hhit : ω∈G.hitEvent o {v | π v<j}) : |
| (G.levelContacts π j o ω).Nonempty := by |
| by_contra h |
| have hd : G.contactDrop π j o ω=∅ := by |
| apply Set.eq_empty_iff_forall_notMem.mpr |
| rintro e ⟨v,hv,w,hw,hend⟩ |
| exact h ⟨v,hv⟩ |
| have hc : Disjoint (G.contactDrop π j o ω) ω := by rw [hd]; exact Set.empty_disjoint _ |
| have he := G.cluster_eq_exterior_of_drop_closed π j o ho hcross ω hc |
| obtain ⟨v,hv,hlt⟩ := hhit |
| have hle : j ≤ π v := G.exteriorCluster_subset_level π j o ho ω (he ▸ hv) |
| exact (not_le.mpr hlt) hle |
|
|
| lemma levelContacts_supply (π : V → ℤ) (j : ℤ) (o : V) (k : ℕ) (ω : Set E) |
| (hne : (G.levelContacts π j o ω).Nonempty) (hfew : ω∉G.fewContacts π j o k) : |
| ∃ S : Finset V, (S : Set V)⊆G.levelContacts π j o ω ∧ k ≤ S.card := by |
| by_cases hfin : (G.levelContacts π j o ω).Finite |
| · refine ⟨hfin.toFinset,by simp,?_⟩ |
| by_contra hcard |
| exact hfew ⟨hfin.toFinset,hfin.coe_toFinset.symm, |
| by simpa using hne,lt_of_not_ge hcard⟩ |
| · obtain ⟨S,hS,hSfin,hcard⟩ := Set.Infinite.exists_subset_ncard_eq hfin k |
| exact ⟨hSfin.toFinset,by simpa using hS,by simpa [Set.ncard_eq_toFinset_card S hSfin] using hcard.ge⟩ |
|
|
| variable [Countable V] [Countable E] |
| def manyContacts (π : V → ℤ) (j : ℤ) (o : V) (k : ℕ) : Set (Set E) := |
| {ω | (G.levelContacts π j o ω).Nonempty}\G.fewContacts π j o k |
|
|
| lemma measurableSet_manyContacts (π : V → ℤ) (j : ℤ) (o : V) (k : ℕ) : |
| MeasurableSet[blockSigma (G.inducedEdges {v | j ≤ π v})] (G.manyContacts π j o k) := by |
| apply MeasurableSet.diff _ (G.measurableSet_fewContacts π j o k) |
| have he : {ω | (G.levelContacts π j o ω).Nonempty}=⋃ v : V, {ω | v∈G.levelContacts π j o ω} := by |
| ext ω |
| simp [Set.Nonempty] |
| rw [he] |
| exact MeasurableSet.iUnion (fun v => |
| @Measurable.setOf (Set E) (blockSigma (G.inducedEdges {v | j ≤ π v})) _ |
| ((measurable_pi_apply v).comp (G.measurable_levelContacts π j o))) |
|
|
| theorem honest_contact_selection (π : V → ℤ) (j : ℤ) (o : V) (k : ℕ) |
| (C : V → V → Prop) (hrefl : ∀ v, C v v) (hsymm : ∀ {v w}, C v w → C w v) |
| (M N : ℕ) (hM : 0 < M) (hk : M*N ≤ k) |
| (hfinite : ∀ v, {w | C v w}.Finite) (hbound : ∀ v, {w | C v w}.ncard ≤ M) |
| (s : V → ℝ) (s₀ K : ℝ) (hK : 0 ≤ K) |
| (hs : ∀ v, π v=j → s₀ ≤ s v ∧ s v ≤ 1) (hN : K ≤ N*s₀) : |
| ∃ d : Set E → Finset V, |
| (∀ U, MeasurableSet[blockSigma (G.inducedEdges {v | j ≤ π v})] {ω | d ω=U}) ∧ |
| ∀ ω∈G.manyContacts π j o k, |
| (d ω : Set V)⊆G.levelContacts π j o ω ∧ |
| (d ω : Set V).Pairwise (fun v w => ¬C v w) ∧ |
| K ≤ ∑ v∈d ω, s v ∧ (∑ v∈d ω, s v) ≤ K+1 := by |
| let P := fun (ω : Set E) (U : Finset V) => |
| (U : Set V)⊆G.levelContacts π j o ω ∧ |
| (U : Set V).Pairwise (fun v w => ¬C v w) ∧ |
| K ≤ ∑ v∈U, s v ∧ (∑ v∈U, s v) ≤ K+1 |
| have hm (U : Finset V) : MeasurableSet[blockSigma (G.inducedEdges {v | j ≤ π v})] {ω | P ω U} := |
| (measurableSet_finset_subset _ (G.measurable_levelContacts π j o) U).inter (MeasurableSet.const _) |
| obtain ⟨d,hd,hgood⟩ := @exists_honest_selector (Set E) (Finset V) |
| (blockSigma (G.inducedEdges {v | j ≤ π v})) inferInstance inferInstance P hm |
| refine ⟨d,hd,?_⟩ |
| intro ω hω |
| apply hgood |
| obtain ⟨S,hS,hcard⟩ := G.levelContacts_supply π j o k ω hω.1 hω.2 |
| have hb : ∀ v∈S, (S.filter (C v)).card ≤ M := by |
| intro v _hv |
| have hh : (S.filter (C v) : Set V)⊆{w | C v w} := by |
| intro w hw |
| exact (Finset.mem_filter.mp hw).2 |
| have hz := (Set.ncard_le_ncard hh (hfinite v)).trans (hbound v) |
| simpa only [Set.ncard_coe_finset] using hz |
| obtain ⟨U,hU,hsep,hlo,hhi⟩ := separated_weighted_trials C hrefl hsymm M N hM S hb |
| (hk.trans hcard) s s₀ K hK (fun v hv => hs v (hS hv).2) hN |
| exact ⟨U,(fun _ hv => hS (hU hv)),hsep,hlo,hhi⟩ |
|
|
| end BondGraph |
| end CriticalPercolation |
| end |
| end |
| end |
|
|
| end OAI |
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|
