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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Criticality.FiniteEnergy
import OAI.Probability.CriticalPercolation.Clusters.InfiniteUniquenessGeometry
import OAI.Probability.CriticalPercolation.Clusters.PercolationTail
namespace OAI
section
section
noncomputable section
open Set MeasureTheory ProbabilityTheory
open scoped Classical
namespace CriticalPercolation.BondGraph
variable {V E : Type*} [Countable V] [Countable E] (G : BondGraph V E)
def atMostOneInfinite : Set (Set E) :=
{ω | ∀ x y, (G.cluster ω x).Infinite → (G.cluster ω y).Infinite →
(G.openGraph ω).Reachable x y}
lemma measurableSet_atMostOneInfinite : MeasurableSet G.atMostOneInfinite := by
have heq : G.atMostOneInfinite = ⋂ x, ⋂ y, (G.distinctInfiniteEvent x y)ᶜ := by
ext ω
simp only [atMostOneInfinite,distinctInfiniteEvent,Set.mem_ofPred_eq,Set.mem_iInter,
Set.mem_compl_iff]
constructor
· intro h x y hh
exact hh.2.2 (h x y hh.1 hh.2.1)
· intro h x y hx hy
by_contra hr
exact h x y ⟨hx,hy,hr⟩
rw [heq]
exact MeasurableSet.iInter fun x => MeasurableSet.iInter fun y =>
(G.measurableSet_distinctInfiniteEvent x y).compl
theorem atMostOneInfinite_ae (hconnected : G.fullGraph.Connected)
(hlocal : G.LocallyFinite) (hquasi : G.QuasiTransitive)
(hg : G.SubexponentialGrowth) (p : unitInterval)
(hp : (p:ℝ) ∈ Set.Ioo 0 1) : ∀ᵐ ω ∂G.law p, ω ∈ G.atMostOneInfinite := by
have hn : ∀ (S : Finset E) (x y : V), ∀ᵐ ω ∂G.law p,
G.fullGraph.Adj x y → (S : Set E) ∪ ω ∉ G.distinctInfiniteEvent x y := by
intro S x y
by_cases hxy : G.fullGraph.Adj x y
· have hz := BernoulliProduct.finite_union_preimage_null (fun _ : E => p)
(fun _ => hp.1) (S : Set E) S.finite_toSet (G.distinctInfiniteEvent x y)
(G.measurableSet_distinctInfiniteEvent x y)
(by rw [BernoulliProduct.law_const];
exact G.adjacent_distinctInfinite_null hconnected hlocal hquasi hg p hp x y hxy)
rw [BernoulliProduct.law_const] at hz
have hh : ∀ᵐ ω ∂G.law p, (S : Set E) ∪ ω ∉ G.distinctInfiniteEvent x y := by
rw [ae_iff]
simp only [not_not]
exact hz
exact hh.mono (fun _ h _ => h)
· exact Filter.Eventually.of_forall (fun _ h => False.elim (hxy h))
have hall := ae_all_iff.mpr (fun S => ae_all_iff.mpr (fun x => ae_all_iff.mpr (hn S x)))
filter_upwards [hall] with ω hω
intro x y hx hy
by_contra hxy
obtain ⟨S,hS,a,b,hab,hh⟩ := G.distinctInfinite_force_adjacent hconnected ω x y ⟨hx,hy,hxy⟩
have hnot := hω hS.toFinset a b hab
exact hnot (by simpa only [Set.Finite.coe_toFinset] using hh)
theorem uniqueInfinite_ae (hconnected : G.fullGraph.Connected)
(hlocal : G.LocallyFinite) (hquasi : G.QuasiTransitive)
(hg : G.SubexponentialGrowth) (p : unitInterval)
(hp : (p:ℝ) ∈ Set.Ioo 0 1) (hpositive : 0 < G.law p G.percolates) :
∀ᵐ ω ∂G.law p, ∃ x, (G.cluster ω x).Infinite ∧
∀ y, (G.cluster ω y).Infinite → (G.openGraph ω).Reachable x y := by
have hex : ∀ᵐ ω ∂G.law p, ω ∈ G.percolates := by
rw [ae_iff]
change G.law p G.percolatesᶜ = 0
rw [measure_compl G.measurableSet_percolates (measure_ne_top _ _),
G.percolates_probability_one p hpositive, measure_univ, tsub_self]
filter_upwards [hex,G.atMostOneInfinite_ae hconnected hlocal hquasi hg p hp] with ω hω hu
obtain ⟨x,hx⟩ := hω
exact ⟨x,hx,fun y hy => hu x y hx hy⟩
end CriticalPercolation.BondGraph
end
end
end
end OAI
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