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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.GraphGeometry.HitEvent
import OAI.Probability.CriticalPercolation.GraphGeometry.OpenGraphIso
namespace OAI
section
section
open scoped BigOperators
open scoped Classical ENNReal
open Set
open Set MeasureTheory ProbabilityTheory Filter
open scoped Classical Topology
open Set MeasureTheory ProbabilityTheory
open scoped Classical
open Filter Topology
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
lemma infiniteEvent_subset_hit_compl (o : V) (K : Set V) (hK : K.Finite) :
G.infiniteEvent o ⊆ G.hitEvent o Kᶜ := by
intro ω hω
exact (hω.sdiff hK).nonempty
lemma infiniteEvent_eq_iInter_hit_compl
(K : Set.FiniteExhaustion (Set.univ : Set V)) (o : V) :
G.infiniteEvent o = ⋂ n, G.hitEvent o (K n)ᶜ := by
ext ω
constructor
· intro hω
exact Set.mem_iInter.mpr fun n => G.infiniteEvent_subset_hit_compl o (K n) (K.finite n) hω
· intro hω
change (G.cluster ω o).Infinite
intro hfin
obtain ⟨n, hn⟩ := finite_subset_exhaustion K hfin (Set.subset_univ _)
obtain ⟨v, hv, hvK⟩ := Set.mem_iInter.mp hω n
exact hvK (hn hv)
lemma tendsto_real_hit_compl [Countable V] [Countable E]
(p : unitInterval) (K : Set.FiniteExhaustion (Set.univ : Set V)) (o : V) :
Tendsto (fun n => (G.law p).real (G.hitEvent o (K n)ᶜ)) atTop
(𝓝 ((G.law p).real (G.infiniteEvent o))) := by
have h := tendsto_measure_iInter_atTop (μ := G.law p)
(fun n => (G.measurableSet_hitEvent o (K n)ᶜ).nullMeasurableSet)
(show Antitone (fun n => G.hitEvent o (K n)ᶜ) from
fun _ _ h => G.hitEvent_mono_target o (Set.compl_subset_compl.mpr (K.mono h)))
⟨0, measure_ne_top _ _⟩
rw [← G.infiniteEvent_eq_iInter_hit_compl K o] at h
exact (ENNReal.tendsto_toReal (measure_ne_top _ _)).comp h
lemma real_hit_singleton_pos (hconnected : G.fullGraph.Connected)
(p : unitInterval) (hp : 0 < (p : ℝ)) (o v : V) :
0 < (G.law p).real (G.hitEvent o {v}) := by
obtain ⟨s, hs, _, hsv⟩ := G.mem_cluster_finite_witness (hconnected o v)
have hsub : {ω : Set E | s ⊆ ω} ⊆ G.hitEvent o {v} := by
intro ω hω
exact ⟨v, G.cluster_mono hω o hsv, rfl⟩
have hpos : 0 < (G.law p).real {ω : Set E | s ⊆ ω} := by
rw [law, ← BernoulliProduct.law_const (E := E) p]
rw [BernoulliProduct.real_law_subset p s hs]
exact pow_pos hp _
exact lt_of_lt_of_le hpos (measureReal_mono hsub)
lemma real_infiniteEvent_transfer [Countable V] [Countable E]
(p : unitInterval) (o v : V) :
(G.law p).real (G.hitEvent o {v}) * (G.law p).real (G.infiniteEvent v) ≤
(G.law p).real (G.infiniteEvent o) := by
let K := (Set.to_countable (Set.univ : Set V)).finiteExhaustion
have hn (n : ℕ) :
(G.law p).real (G.hitEvent o {v}) * (G.law p).real (G.infiniteEvent v) ≤
(G.law p).real (G.hitEvent o (K n)ᶜ) := by
apply le_trans (G.joint_gluing p o {v} (K n)ᶜ
((G.law p).real (G.infiniteEvent v)) ?_)
(measureReal_mono Set.inter_subset_right)
intro w hw
rcases Set.mem_singleton_iff.mp hw with rfl
exact measureReal_mono (G.infiniteEvent_subset_hit_compl w (K n) (K.finite n))
exact ge_of_tendsto (G.tendsto_real_hit_compl p K o) (Eventually.of_forall hn)
lemma real_infiniteEvent_pos_all [Countable V] [Countable E]
(hconnected : G.fullGraph.Connected) (p : unitInterval) (hp : 0 < (p : ℝ))
{v : V} (hv : 0 < (G.law p).real (G.infiniteEvent v)) (o : V) :
0 < (G.law p).real (G.infiniteEvent o) :=
lt_of_lt_of_le (mul_pos (G.real_hit_singleton_pos hconnected p hp o v) hv)
(G.real_infiniteEvent_transfer p o v)
theorem uniform_rooted_probability
(hconnected : G.fullGraph.Connected) (hlocal : G.LocallyFinite)
(hquasi : G.QuasiTransitive) (p : unitInterval)
(hp : 0 < G.law p G.percolates) :
∃ θ : ℝ, 0 < θ ∧ ∀ o : V, θ ≤ (G.law p).real (G.infiniteEvent o) := by
let := G.countable_vertices hconnected hlocal
let := G.countable_bonds hconnected hlocal
have hp0 : 0 < (p : ℝ) := by
by_contra h
have hz : p = 0 := Subtype.ext (le_antisymm (not_lt.mp h) p.property.1)
subst p
rw [G.law_zero_percolates] at hp
exact (lt_irrefl _ hp)
have hn : ¬ ∀ v, G.law p (G.infiniteEvent v) = 0 := by
intro h
exact (ne_of_gt hp) ((G.percolates_null_iff _).mpr h)
push Not at hn
obtain ⟨v, hv⟩ := hn
have hvpos : 0 < (G.law p).real (G.infiniteEvent v) :=
ENNReal.toReal_pos hv (measure_ne_top _ _)
have hall := G.real_infiniteEvent_pos_all hconnected p hp0 hvpos
obtain ⟨S, hS, horbit⟩ := hquasi
have hSne : S.Nonempty := by
obtain ⟨s, hs, _⟩ := horbit v
exact ⟨s, hs⟩
obtain ⟨s, hs, hmin⟩ := Set.exists_min_image S
(fun o => (G.law p).real (G.infiniteEvent o)) hS hSne
refine ⟨(G.law p).real (G.infiniteEvent s), hall s, ?_⟩
intro o
obtain ⟨t, ht, φ, rfl⟩ := horbit o
rw [Aut.infiniteEvent_real]
exact hmin t ht
end CriticalPercolation.BondGraph
open Set MeasureTheory ProbabilityTheory Filter
open scoped Topology Classical
end
end
end OAI
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