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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.Gluing.SelectedFresh
import OAI.Probability.CriticalPercolation.ProductMeasure.LevelContactMeasurability
import OAI.Probability.CriticalPercolation.Criticality.FewOpenBonds
import OAI.Probability.CriticalPercolation.Exploration.FrontierDivergence
namespace OAI
section
noncomputable section
open Set MeasureTheory ProbabilityTheory Filter
open scoped Classical ENNReal Topology
namespace CriticalPercolation.BernoulliProduct
variable {E : Type*} [Countable E]
lemma measurableSet_fewOpenBonds (S : Set E) (N : ℕ) :
MeasurableSet[blockSigma S] {ω : Set E | (S∩ω).ncard ≤ N} := by
simpa only [Set.inter_comm,Set.preimage_ofPred_eq,Set.Iic,Function.comp_apply] using
(measurableSet_Iic.preimage (measurable_ncard.comp (measurable_inter_block S)))
end CriticalPercolation.BernoulliProduct
namespace CriticalPercolation.BondGraph
open BernoulliProduct
variable {V E : Type*} [Countable V] [Countable E] (G : BondGraph V E)
def fewOpenFreshBoundary (X : Set E) (v : V) (N : ℕ) : Set (Set E) :=
{ω | (G.cluster (ω∩X) v).Finite ∧ (G.freshClusterBoundary X v ω ∩ ω).ncard ≤ N}
lemma fewOpenFreshBoundary_probability_le (hl : G.LocallyFinite) (p : unitInterval)
(X : Set E) (v : V) (N L : ℕ) :
(G.law p).real (G.infiniteEvent v ∩ G.fewOpenFreshBoundary X v N) ≤
(G.law p).real (G.infiniteEvent v ∩ G.smallFreshBoundary X v ((N+1)*L)) +
(N+1 : ℝ)*(1-(p:ℝ))^L := by
let M := (N+1)*L
let C : Set (Set E) := {ω | (G.cluster (ω∩X) v).Finite ∧
M ≤ (G.freshClusterBoundary X v ω).ncard}
let d : Set E → Finset E := fun ω => finiteSetChoice (G.freshClusterBoundary X v ω)
let A : Finset E → Set (Set E) := fun S =>
{ω | (((S : Set E)∩Xᶜ)∩ω).ncard ≤ N}
have hC : MeasurableSet[blockSigma X] C := 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_Ici.preimage
(measurable_ncard.comp (G.measurable_freshClusterBoundary X v)))
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_fewOpenBonds _ N)
have he (ω : Set E) (hω : (G.cluster (ω∩X) v).Finite) :
((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).subset Set.sdiff_subset
rw [finiteSetChoice_coe hf]
exact Set.inter_eq_left.mpr (Set.sdiff_subset_compl _ _)
let r : ℝ := (N+1 : ℝ)*(1-(p:ℝ))^L
have hr : 0 ≤ r := mul_nonneg (by positivity) (pow_nonneg (sub_nonneg.mpr p.2.2) _)
have hb (ω : Set E) (hω : ω∈C) : (BernoulliProduct.law (fun _ : E => p)).real (A (d ω)) ≤ r := by
change (BernoulliProduct.law (fun _ : E => p)).real
{τ | (((d ω : Set E)∩Xᶜ)∩τ).ncard ≤ N} ≤ _
rw [he ω hω.1]
exact fewOpenBonds_probability_le p _
((G.finite_cutEdges hl hω.1).subset Set.sdiff_subset) N L hω.2
have ht := selected_fresh_bound (fun _ : E => p) X Xᶜ disjoint_compl_right
d A hd hA C hC r hr hb
simp only [law_const] at ht
change (G.law p).real (C∩selectedEvent d A) ≤ r*(G.law p).real C at ht
have ht' : (G.law p).real (C∩selectedEvent d A) ≤ r :=
ht.trans ((mul_le_mul_of_nonneg_left measureReal_le_one hr).trans_eq (mul_one r))
have hsub : G.infiniteEvent v ∩ G.fewOpenFreshBoundary X v N ⊆
(G.infiniteEvent v ∩ G.smallFreshBoundary X v M) ∪ (C∩selectedEvent d A) := by
intro ω hω
by_cases h : (G.freshClusterBoundary X v ω).ncard ≤ M
· exact Or.inl ⟨hω.1,hω.2.1,h⟩
· right
refine ⟨⟨hω.2.1,(Nat.le_of_lt (Nat.lt_of_not_ge h))⟩,?_⟩
change (((d ω : Set E)∩Xᶜ)∩ω).ncard ≤ N
rw [he ω hω.2.1]
exact hω.2.2
exact ((measureReal_mono hsub).trans (measureReal_union_le _ _)).trans
(add_le_add_right ht' _)
theorem fewOpenFreshBoundary_probability_tendsto_zero
(hl : G.LocallyFinite) (p : unitInterval) (hp : 0 < (p:ℝ)) (hp1 : (p:ℝ)<1)
(X : ℕ → Set E) (hX : ∀ S : Finset E, ∀ᶠ n in atTop, (S : Set E) ⊆ X n)
(v : V) (N : ℕ) :
Tendsto (fun n => G.law p (G.infiniteEvent v ∩ G.fewOpenFreshBoundary (X n) v N))
atTop (𝓝 0) := by
have hpow : Tendsto (fun L : ℕ => (N+1 : ℝ)*(1-(p:ℝ))^L) atTop (𝓝 0) := by
simpa only [mul_zero] using (tendsto_pow_atTop_nhds_zero_of_lt_one
(sub_nonneg.mpr p.2.2) (by linarith : 1-(p:ℝ)<1)).const_mul (N+1 : ℝ)
have ht : Tendsto (fun n => (G.law p).real
(G.infiniteEvent v ∩ G.fewOpenFreshBoundary (X n) v N)) atTop (𝓝 0) := by
apply tendsto_order.mpr
constructor
· intro a ha
exact Eventually.of_forall (fun _ => lt_of_lt_of_le ha measureReal_nonneg)
· intro b hb
obtain ⟨L,hL⟩ := (hpow.eventually (gt_mem_nhds (half_pos hb))).exists
have hs := G.smallFreshBoundary_probability_tendsto_zero hl p hp1 X hX v ((N+1)*L)
have hsreal := ENNReal.tendsto_toReal ENNReal.zero_ne_top |>.comp hs
simp only [ENNReal.toReal_zero,Function.comp_def] at hsreal
change Tendsto (fun n => (G.law p).real
(G.infiniteEvent v ∩ G.smallFreshBoundary (X n) v ((N+1)*L))) atTop (𝓝 0) at hsreal
filter_upwards [hsreal.eventually (gt_mem_nhds (half_pos hb))] with n hn
exact lt_of_le_of_lt (G.fewOpenFreshBoundary_probability_le hl p (X n) v N L)
(by linarith)
have hh := ENNReal.tendsto_ofReal ht
simpa only [ENNReal.ofReal_zero,Measure.real,ENNReal.ofReal_toReal (measure_ne_top _ _)] using hh
end CriticalPercolation.BondGraph
end
end
end OAI
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