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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 | 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
end OAI
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