| import Mathlib |
| import OAI.Probability.CriticalPercolation.Criticality.InfiniteEscapeLowerBound |
|
|
| namespace OAI |
|
|
| section |
| section CurrentModule9 |
| noncomputable section |
| open Set MeasureTheory |
| open scoped Classical BigOperators |
|
|
| namespace CriticalPercolation.Sharpness |
| open ClusterDeletion |
|
|
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| lemma measurableSet_boundary (hl : G.LocallyFinite) |
| (o : V) (S : Set V) (hS : S.Finite) (e : E) : |
| MeasurableSet {ω | boundaryReach G Sᶜ (delete (G.openGraph ω) Sᶜ) o e} := by |
| let K := G.incidentEdges S |
| have hK : K.Finite := finite_incident G hl hS |
| let := hK.fintype |
| have hm : Measurable (fun ω : Set E => (Subtype.val ⁻¹' ω : Set K)) := |
| Measurable.of_eval (fun edge => measurable_set_mem edge.val) |
| have he : {ω | boundaryReach G Sᶜ (delete (G.openGraph ω) Sᶜ) o e} = |
| (fun ω : Set E => (Subtype.val ⁻¹' ω : Set K)) ⁻¹' |
| {η | boundaryReach G Sᶜ |
| (delete ((G.restrictEdges K).openGraph η) Sᶜ) o e} := by |
| ext ω |
| simp only [Set.mem_preimage,Set.mem_ofPred_eq,G.restrictEdges_openGraph, |
| delete_restrict_incident G (K := K) Subset.rfl] |
| rw [he] |
| exact hm (.of_discrete) |
|
|
| noncomputable def deleteImageIso (φ : G.Aut) (ω : Set E) (S : Set V) : |
| delete (G.openGraph ω) S ≃g delete (G.openGraph (φ.edge '' ω)) (φ.vertex '' S) where |
| toEquiv := φ.vertex |
| map_rel_iff' := by |
| intro x y |
| change ((G.openGraph (φ.edge '' ω)).Adj (φ.vertex x) (φ.vertex y) ∧ |
| φ.vertex x ∉ φ.vertex '' S ∧ φ.vertex y ∉ φ.vertex '' S) ↔ _ |
| simp only [BondGraph.Aut.adj_map,Set.mem_image,φ.vertex.injective.eq_iff,exists_eq_right] |
| rfl |
|
|
| lemma boundary_image (φ : G.Aut) (S : Set V) (o : V) (e : E) (ω : Set E) : |
| boundaryReach G (φ.vertex '' S)ᶜ |
| (delete (G.openGraph (φ.edge '' ω)) (φ.vertex '' S)ᶜ) (φ.vertex o) (φ.edge e) ↔ |
| boundaryReach G Sᶜ (delete (G.openGraph ω) Sᶜ) o e := by |
| have himg : (φ.vertex '' S)ᶜ = φ.vertex '' Sᶜ := by |
| ext v |
| obtain ⟨x,rfl⟩ := φ.vertex.surjective v |
| simp only [Set.mem_compl_iff,Set.mem_image,φ.vertex.injective.eq_iff,exists_eq_right] |
| rw [himg] |
| constructor |
| · rintro ⟨a,b,he,ha,hb,hr⟩ |
| obtain ⟨x,rfl⟩ := φ.vertex.surjective a |
| obtain ⟨y,rfl⟩ := φ.vertex.surjective b |
| have hx : x ∉ Sᶜ := by simpa only [Set.mem_image,φ.vertex.injective.eq_iff,exists_eq_right] using ha |
| have hy : y ∈ Sᶜ := by simpa only [Set.mem_image,φ.vertex.injective.eq_iff,exists_eq_right] using hb |
| rw [φ.preserves,← Sym2.map_mk] at he |
| exact ⟨x,y,Sym2.map.injective φ.vertex.injective he,hx,hy, |
| (SimpleGraph.Iso.reachable_iff (φ := deleteImageIso G φ ω Sᶜ)).mp hr⟩ |
| · rintro ⟨x,y,he,hx,hy,hr⟩ |
| refine ⟨φ.vertex x,φ.vertex y,?_,?_,?_,?_⟩ |
| · rw [φ.preserves,he,Sym2.map_mk] |
| · simpa only [Set.mem_image,φ.vertex.injective.eq_iff,exists_eq_right] using hx |
| · exact ⟨y,hy,rfl⟩ |
| · exact (SimpleGraph.Iso.reachable_iff (φ := deleteImageIso G φ ω Sᶜ)).mpr hr |
|
|
| lemma boundary_real_image (hl : G.LocallyFinite) (φ : G.Aut) (p : unitInterval) |
| (S : Set V) (hS : S.Finite) (o : V) (e : E) : |
| (G.law p).real {ω | boundaryReach G (φ.vertex '' S)ᶜ |
| (delete (G.openGraph ω) (φ.vertex '' S)ᶜ) (φ.vertex o) (φ.edge e)} = |
| (G.law p).real {ω | boundaryReach G Sᶜ (delete (G.openGraph ω) Sᶜ) o e} := by |
| change (G.law p _).toReal = (G.law p _).toReal |
| congr 1 |
| calc |
| _ = (G.law p).map (fun ω : Set E => φ.edge '' ω) |
| {ω | boundaryReach G (φ.vertex '' S)ᶜ |
| (delete (G.openGraph ω) (φ.vertex '' S)ᶜ) (φ.vertex o) (φ.edge e)} := by |
| rw [G.law_map_aut p φ] |
| _ = _ := by |
| rw [Measure.map_apply (BernoulliProduct.measurable_image_equiv φ.edge) |
| (measurableSet_boundary G hl _ _ (hS.image _) _)] |
| congr 1 |
| ext ω |
| exact boundary_image G φ S o e ω |
|
|
| lemma phi_image (hl : G.LocallyFinite) (φ : G.Aut) (p : unitInterval) |
| (S : Set V) (hS : S.Finite) (o : V) : |
| phi G p (φ.vertex o) (φ.vertex '' S) = phi G p o S := by |
| unfold phi |
| congr 1 |
| rw [← φ.edge.tsum_eq] |
| apply tsum_congr |
| intro e |
| exact boundary_real_image G hl φ p S hS o e |
|
|
| theorem uniform_subcritical_domains [Countable V] [Countable E] [Nonempty V] |
| (hl : G.LocallyFinite) (hq : G.QuasiTransitive) |
| (p : unitInterval) (hp : p < G.criticalProbability) : |
| ∃ N : ℕ, ∃ a : ℝ, a < 1 ∧ ∀ o : V, |
| ∃ S : Set V, S.Finite ∧ o ∈ S ∧ S.ncard ≤ N ∧ phi G p o S ≤ a := by |
| obtain ⟨R,hR,horbit⟩ := hq |
| have hne : R.Nonempty := by |
| obtain ⟨v⟩ := ‹Nonempty V› |
| obtain ⟨r,hr,_⟩ := horbit v |
| exact ⟨r,hr⟩ |
| choose D hD hoD hphi using subcritical_finite_size G hl p hp |
| obtain ⟨r,hr,hmax⟩ := Set.exists_max_image R (fun o => phi G p o (D o)) hR hne |
| let N := ∑ o ∈ hR.toFinset, (D o).ncard |
| refine ⟨N,phi G p r (D r),hphi r,?_⟩ |
| intro o |
| obtain ⟨s,hs,φ,rfl⟩ := horbit o |
| refine ⟨φ.vertex '' D s,(hD s).image _,⟨s,hoD s,rfl⟩,?_,?_⟩ |
| · rw [Set.ncard_image_of_injective _ φ.vertex.injective] |
| exact Finset.single_le_sum (f := fun o => (D o).ncard) |
| (fun _ _ => Nat.zero_le _) (hR.mem_toFinset.mpr hs) |
| · rw [phi_image G hl φ p (D s) (hD s) s] |
| exact hmax s hs |
|
|
| end CriticalPercolation.Sharpness |
| end |
| end CurrentModule9 |
|
|
| end |
|
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| end OAI |
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