| import Mathlib |
| import OAI.Probability.CriticalPercolation.GraphGeometry.OpenGraphIso |
| import OAI.Probability.CriticalPercolation.Criticality.FiniteEnergy |
| import OAI.Probability.CriticalPercolation.Clusters.FiniteClusterCover |
| import OAI.Probability.CriticalPercolation.Clusters.GraphErgodicity |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set MeasureTheory |
| open scoped Classical |
|
|
| namespace CriticalPercolation.BondGraph |
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| lemma Aut.atMostOneInfinite_image_iff (φ : G.Aut) (ω : Set E) : |
| φ.edge '' ω ∈ G.atMostOneInfinite ↔ ω ∈ G.atMostOneInfinite := by |
| have hi (x : V) : (G.cluster (φ.edge '' ω) (φ.vertex x)).Infinite ↔ |
| (G.cluster ω x).Infinite := by |
| rw [←φ.cluster_image] |
| exact Set.infinite_image_iff φ.vertex.injective.injOn |
| have hr (x y : V) := SimpleGraph.Iso.reachable_iff (φ := Aut.openGraphIso G φ ω) |
| (u := x) (v := y) |
| constructor |
| · intro h x y hx hy |
| exact (hr x y).mp (h (φ.vertex x) (φ.vertex y) ((hi x).mpr hx) ((hi y).mpr hy)) |
| · intro h x y hx hy |
| obtain ⟨a,rfl⟩ := φ.vertex.surjective x |
| obtain ⟨b,rfl⟩ := φ.vertex.surjective y |
| exact (hr a b).mpr (h a b ((hi a).mp hx) ((hi b).mp hy)) |
|
|
| lemma atMostOneInfinite_zero_one [Infinite V] [Countable V] [Countable E] |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive) |
| (p : unitInterval) : G.law p G.atMostOneInfinite = 0 ∨ |
| G.law p G.atMostOneInfinite = 1 := |
| G.invariant_event_zero_one hc hl hq p G.atMostOneInfinite G.measurableSet_atMostOneInfinite |
| (fun φ => Set.ext fun ω => Aut.atMostOneInfinite_image_iff G φ ω) |
|
|
| lemma no_finite_infiniteClusterCover_ae_of_unique_null [Countable V] [Countable E] |
| (hc : G.fullGraph.Connected) (p : unitInterval) (hp : (0:ℝ) < p) |
| (hnull : G.law p G.atMostOneInfinite = 0) : |
| ∀ᵐ ω ∂G.law p, ∀ S : Finset V, ¬ G.infiniteClusterCover ω (S : Set V) := by |
| apply ae_all_iff.mpr |
| intro S |
| obtain ⟨o⟩ := hc.nonempty |
| obtain ⟨J,hJ,hconn⟩ := G.finite_bonds_connect_set hc o (S : Set V) S.finite_toSet |
| have hn := BernoulliProduct.finite_union_preimage_null (fun _ : E => p) (fun _ => hp) |
| J hJ G.atMostOneInfinite G.measurableSet_atMostOneInfinite |
| (by simpa only [BernoulliProduct.law_const,law] using hnull) |
| have hn' : G.law p ((fun ω : Set E => J ∪ ω) ⁻¹' G.atMostOneInfinite) = 0 := by |
| simpa only [BernoulliProduct.law_const,law] using hn |
| rw [ae_iff] |
| apply measure_mono_null _ hn' |
| intro ω hω |
| change J ∪ ω ∈ G.atMostOneInfinite |
| rw [Set.union_comm] |
| exact G.atMostOneInfinite_union_of_cover ω J hJ o (S : Set V) hconn (not_not.mp hω) |
| |
| def infinitelyManyInfiniteClusters (ω : Set E) : Prop := |
| ∀ S : Finset V, ∃ v, (G.cluster ω v).Infinite ∧ |
| ∀ s ∈ S, ¬ (G.openGraph ω).Reachable v s |
| |
| lemma infinitelyManyInfiniteClusters_iff_no_finite_cover (ω : Set E) : |
| G.infinitelyManyInfiniteClusters ω ↔ |
| ∀ S : Finset V, ¬ G.infiniteClusterCover ω (S : Set V) := by |
| simp only [infinitelyManyInfiniteClusters,infiniteClusterCover] |
| push Not |
| rfl |
| |
| theorem unique_or_infinitely_many_ae [Infinite V] |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive) |
| (p : unitInterval) (hp : (0:ℝ) < p) : |
| (∀ᵐ ω ∂G.law p, ω ∈ G.atMostOneInfinite) ∨ |
| (∀ᵐ ω ∂G.law p, G.infinitelyManyInfiniteClusters ω) := by |
| let := G.countable_vertices hc hl |
| let := G.countable_bonds hc hl |
| rcases G.atMostOneInfinite_zero_one hc hl hq p with hz | ho |
| · right |
| filter_upwards [G.no_finite_infiniteClusterCover_ae_of_unique_null hc p hp hz] with ω hω |
| exact (G.infinitelyManyInfiniteClusters_iff_no_finite_cover ω).mpr hω |
| · left |
| rw [ae_iff] |
| change G.law p G.atMostOneInfiniteᶜ = 0 |
| rw [measure_compl G.measurableSet_atMostOneInfinite (measure_ne_top _ _),ho, |
| measure_univ,tsub_self] |
| |
| end CriticalPercolation.BondGraph |
| end |
| end |
| |
| end OAI |
| |
