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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 | 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
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