| import Mathlib |
| import OAI.Probability.CriticalPercolation.Model |
| import OAI.Probability.CriticalPercolation.GraphGeometry.FiniteInduced |
| import OAI.Probability.CriticalPercolation.Gluing.ConditionalGluing |
| import OAI.Probability.CriticalPercolation.Gluing.ContactSelection |
| import OAI.Probability.CriticalPercolation.Gluing.FewContacts |
| import OAI.Probability.CriticalPercolation.Gluing.FreshContactTransfer |
| import OAI.Probability.CriticalPercolation.Gluing.FreshTrial |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open Set Function MeasureTheory ProbabilityTheory |
| open scoped Classical BigOperators ENNReal |
|
|
| namespace CriticalPercolation.BondGraph |
| open BernoulliProduct |
|
|
| variable [Countable V] [Countable E] (G : BondGraph V E) |
|
|
| theorem fresh_collar_transfer_from_geometry (hlocal : G.LocallyFinite) |
| (q : E → unitInterval) (π : V → ℤ) (o : V) (Input T : Set V) |
| (J : Finset ℤ) (hJ : J.Nonempty) (ho : ∀ j∈J, j ≤ π o) |
| (hInput : ∀ j∈J, ∀ v∈Input, π v<j) |
| (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 M N : ℕ) (hD : ∀ j∈J, ∀ v, π v=j → (G.incidentEdges {v}).ncard ≤ D) |
| (hM : 0 < M) (hk : M*N ≤ k) |
| (C : V → V → Prop) (hrefl : ∀ v, C v v) (hsymm : ∀ {v w}, C v w → C w v) |
| (hfinite : ∀ v, {w | C v w}.Finite) (hbound : ∀ v, {w | C v w}.ncard ≤ M) |
| (F : ℤ → V → Set E) (B : ℤ → V → Set V) |
| (hF : ∀ j∈J, ∀ v, F j v ⊆ (G.inducedEdges {w | j ≤ π w})ᶜ) |
| (hconnect : ∀ j∈J, ∀ v, π v=j → ∀ b∈B j v, (G.openGraph (F j v)).Reachable v b) |
| (hdis : ∀ j∈J, ∀ v w, ¬C v w → Disjoint (F j v) (F j w)) |
| (s₀ K η lam : ℝ) (hK : 0 ≤ K) (hη : 0 ≤ η) (hlam : 0 < lam) |
| (hs : ∀ j∈J, ∀ v, π v=j → s₀ ≤ (BernoulliProduct.law q).real (allOpen (F j v))) |
| (hN : K ≤ N*s₀) |
| (hmean : ∀ j∈J, ∀ v, π v=j → |
| 1-η ≤ (BernoulliProduct.law q).real (G.seedHitEvent (B j v) T)) : |
| (BernoulliProduct.law q).real (G.hitEvent o Input\G.hitEvent o T) ≤ |
| 1/((J.card : ℝ)*(1-(p : ℝ))^(D*k))+Real.exp (-K)+(K+1)*η/lam+lam := by |
| obtain ⟨j,hj,hfew⟩ := G.exists_few_contact_level hlocal q π o J hJ ho hcross p hp hq D k hD |
| obtain ⟨d,hd,hsel⟩ := G.honest_contact_selection π j o k C hrefl hsymm M N hM hk |
| hfinite hbound (fun v => (BernoulliProduct.law q).real (allOpen (F j v))) s₀ K hK |
| (fun v hv => ⟨hs j hj v hv,measureReal_le_one⟩) hN |
| have hh := G.fresh_contact_transfer q (G.inducedEdges ) |
| (G.inducedEdges )ᶜ disjoint_compl_right d (F j) (B j) |
| T Input o hd (hF j hj) (hconnect j hj) |
| (G.manyContacts π j o k) (G.fewContacts π j o k) (G.measurableSet_manyContacts π j o k) |
| (by |
| intro ω hω |
| have hne := G.levelContacts_nonempty_of_hit π j o (ho j hj) (hcross j hj) ω |
| (G.hitEvent_mono_target o (hInput j hj) hω) |
| by_cases hf : ω∈G.fewContacts π j o k |
| · exact Or.inl hf |
| · exact Or.inr ⟨hne,hf⟩) |
| (by |
| intro ω hω v hv |
| have hmem := (hsel ω hω).1 hv |
| exact ⟨hmem.2,G.levelContacts_reachable π j o ω hmem⟩) |
| K η lam (by linarith) hη hlam (hmean j hj) |
| (by |
| intro ω hω v hv w hw hvw |
| exact hdis j hj v w ((hsel ω hω).2.1 hv hw hvw)) |
| (fun ω hω => (hsel ω hω).2.2.1) (fun ω hω => (hsel ω hω).2.2.2) |
| linarith |
|
|
| end CriticalPercolation.BondGraph |
| end |
| end |
| end |
|
|
| end OAI |
|
|
