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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 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphGeometry.CutEdges
import OAI.Probability.CriticalPercolation.GraphGeometry.OpenGraphIso
import OAI.Probability.CriticalPercolation.ProductMeasure.LevelCutProbability
import OAI.Probability.CriticalPercolation.ProductMeasure.LevelContactMeasurability
namespace OAI
section
noncomputable section
open Set MeasureTheory ProbabilityTheory Filter
open scoped Classical ENNReal symmDiff Topology
namespace CriticalPercolation.BondGraph
open BernoulliProduct
variable {V E : Type*} (G : BondGraph V E)
def freshClusterBoundary (X : Set E) (v : V) (ω : Set E) : Set E :=
G.cutEdges (G.cluster (ω∩X) v) \ X
lemma cutEdges_disjoint_cluster (ω : Set E) (v : V) :
Disjoint (G.cutEdges (G.cluster ω v)) ω := by
apply Set.disjoint_left.mpr
rintro e ⟨x,y,he,hx,hy⟩ hω
have hne : x ≠ y := fun h => hx (h ▸ hy)
exact hx (hy.trans (SimpleGraph.Adj.reachable
(show (G.openGraph ω).Adj x y from ⟨⟨e,hω,he⟩,hne⟩)).symm)
lemma finite_cutEdges (hl : G.LocallyFinite) {S : Set V} (hS : S.Finite) :
(G.cutEdges S).Finite := by
apply (G.finite_incident hl hS).subset
rintro e ⟨x,y,he,_,hy⟩
exact ⟨y,hy,by rw [he]; simp⟩
lemma cluster_eq_restricted_of_fresh_closed (X : Set E) (v : V) (ω : Set E)
(h : Disjoint (G.freshClusterBoundary X v ω) ω) :
G.cluster ω v = G.cluster (ω∩X) v := by
apply Set.Subset.antisymm
· intro x hx
by_contra hn
apply G.not_reachable_of_cut_closed (G.cluster (ω∩X) v) ω ?_ hn
(SimpleGraph.Reachable.refl v) hx.symm
intro e he hω
by_cases hX : e∈X
· exact Set.disjoint_left.mp (G.cutEdges_disjoint_cluster (ω∩X) v) he ⟨hω,hX⟩
· exact Set.disjoint_left.mp h ⟨he,hX⟩ hω
· exact G.cluster_mono Set.inter_subset_left v
lemma measurable_freshClusterBoundary [Countable V] [Countable E] (X : Set E) (v : V) :
@Measurable (Set E) (Set E) (blockSigma X) inferInstance (G.freshClusterBoundary X v) := by
let : MeasurableSpace (Set E) := blockSigma X
refine Measurable.of_eval ?_
intro e
exact @Measurable.and (Set E) (blockSigma X) _ _
(.exists fun x => .exists fun y => measurable_const.and
((((G.measurable_reachable v x).comp (measurable_inter_block X)).not).and
((G.measurable_reachable v y).comp (measurable_inter_block X)))) measurable_const
def smallFreshBoundary (X : Set E) (v : V) (M : ℕ) : Set (Set E) :=
{ω | (G.cluster (ω∩X) v).Finite ∧ (G.freshClusterBoundary X v ω).ncard ≤ M}
lemma measurableSet_smallFreshBoundary [Countable V] [Countable E] (X : Set E) (v : V) (M : ℕ) :
MeasurableSet[blockSigma X] (G.smallFreshBoundary X v M) := by
have hf : MeasurableSet[blockSigma X] {ω | (G.cluster (ω∩X) v).Finite} := by
let : MeasurableSpace (Set E) := blockSigma X
simpa only [Function.comp_apply,Set.not_infinite] using
(((G.measurable_infinite_cluster v).comp (measurable_inter_block X)).not).setOf
exact hf.inter (measurableSet_Iic.preimage
(measurable_ncard.comp (G.measurable_freshClusterBoundary X v)))
theorem smallFreshBoundary_cost [Countable V] [Countable E]
(hl : G.LocallyFinite) (p : unitInterval) (X : Set E) (v : V) (M : ℕ)
(H : Set (Set E)) (hH : MeasurableSet[blockSigma X] H) :
(1-(p:ℝ))^M * (G.law p).real (H ∩ G.smallFreshBoundary X v M) ≤
(G.law p).real (H \ G.infiniteEvent v) := by
let d : Set E → Finset E := fun ω => finiteSetChoice (G.freshClusterBoundary X v ω)
let A : Finset E → Set (Set E) := fun S => closedBlock ((S : Set E)∩Xᶜ)
have hd (S : Finset E) : MeasurableSet[blockSigma X] {ω | d ω=S} :=
@measurableSet_finiteSetChoice (Set E) E (blockSigma X) inferInstance
(G.freshClusterBoundary X v) (G.measurable_freshClusterBoundary X v) S
have hA (S : Finset E) : MeasurableSet[blockSigma Xᶜ] (A S) :=
blockSigma_mono Set.inter_subset_right _
(measurableSet_closedBlock _ ((S.finite_toSet.subset Set.inter_subset_left).countable))
have he (ω : Set E) (hω : ω∈G.smallFreshBoundary X v M) :
((d ω : Set E)∩Xᶜ)=G.freshClusterBoundary X v ω := by
change (finiteSetChoice (G.freshClusterBoundary X v ω) : Set E)∩Xᶜ = _
have hf : (G.freshClusterBoundary X v ω).Finite :=
(G.finite_cutEdges hl hω.1).subset Set.sdiff_subset
rw [finiteSetChoice_coe hf]
exact Set.inter_eq_left.mpr (Set.sdiff_subset_compl _ _)
have hb (ω : Set E) (hω : ω∈H∩G.smallFreshBoundary X v M) :
(1-(p:ℝ))^M ≤ (G.law p).real (A (d ω)) := by
change _ ≤ (G.law p).real (closedBlock ((d ω : Set E)∩Xᶜ))
rw [he ω hω.2]
simpa only [law,←law_const,freshClusterBoundary] using law_closedBlock_lower (fun _ : E => p) _
((G.finite_cutEdges hl hω.2.1).subset Set.sdiff_subset) p M hω.2.2 (fun _ _ => le_rfl)
have hh := selected_fresh_lower (fun _ : E => p) X Xᶜ disjoint_compl_right d A hd hA
(H∩G.smallFreshBoundary X v M) (hH.inter (G.measurableSet_smallFreshBoundary X v M))
((1-(p:ℝ))^M) (pow_nonneg (sub_nonneg.mpr p.2.2) _) (by
simpa only [law,←law_const] using hb)
simp only [law_const] at hh
change (1-(p:ℝ))^M * (G.law p).real (H ∩ G.smallFreshBoundary X v M) ≤
(G.law p).real (H ∩ G.smallFreshBoundary X v M ∩ selectedEvent d A) at hh
apply hh.trans
apply measureReal_mono (h₂ := measure_ne_top _ _)
rintro ω ⟨⟨hωH,hω⟩,hclosed⟩
refine ⟨hωH,?_⟩
have hc : Disjoint (G.freshClusterBoundary X v ω) ω := by
change Disjoint ((d ω : Set E)∩Xᶜ) ω at hclosed
rwa [he ω hω] at hclosed
rw [infiniteEvent,Set.mem_ofPred_eq,G.cluster_eq_restricted_of_fresh_closed X v ω hc]
exact hω.1.not_infinite
end CriticalPercolation.BondGraph
end
end
end OAI
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