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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Model
import OAI.Probability.CriticalPercolation.Criticality.CylinderApproximation
import OAI.Probability.CriticalPercolation.Exploration.SliceExploration
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*} [Countable V] [Countable E] (G : BondGraph V E)
lemma smallFreshBoundary_approx_bound (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 (G.infiniteEvent v ∩ G.smallFreshBoundary X v M) ≤
(1 + (1-(p:ℝ))^M) * (G.law p).real (H ∆ G.infiniteEvent v) := by
have hcost := G.smallFreshBoundary_cost hl p X v M H hH
have hcover : G.infiniteEvent v ∩ G.smallFreshBoundary X v M ⊆
(H ∩ G.smallFreshBoundary X v M) ∪ (H ∆ G.infiniteEvent v) := by
intro ω hω
by_cases h : ω∈H
· exact Or.inl ⟨h,hω.2⟩
· exact Or.inr (Or.inr ⟨hω.1,h⟩)
have hm := (measureReal_mono hcover (μ := G.law p)).trans
(measureReal_union_le _ _)
have he : (G.law p).real (H \ G.infiniteEvent v) ≤
(G.law p).real (H ∆ G.infiniteEvent v) :=
measureReal_mono (by intro ω hω; exact Or.inl hω)
have hr : 0 ≤ (1-(p:ℝ))^M := pow_nonneg (sub_nonneg.mpr p.2.2) _
nlinarith only [hcost,hm,he,hr]
theorem smallFreshBoundary_probability_tendsto_zero
(hl : G.LocallyFinite) (p : unitInterval) (hp : (p:ℝ)<1)
(X : ℕ → Set E) (hX : ∀ S : Finset E, ∀ᶠ n in atTop, (S : Set E) ⊆ X n)
(v : V) (M : ℕ) :
Tendsto (fun n => G.law p (G.infiniteEvent v ∩ G.smallFreshBoundary (X n) v M))
atTop (𝓝 0) := by
let r : ℝ := (1-(p:ℝ))^M
have hr : 0 < r := pow_pos (sub_pos.mpr hp) _
have ht : Tendsto (fun n => (G.law p).real
(G.infiniteEvent v ∩ G.smallFreshBoundary (X n) v M)) 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
have hδ : 0 < b*r/(1+r) := div_pos (mul_pos hb hr) (by linarith)
obtain ⟨S,H,hH,happrox⟩ := exists_finiteCoordinate_approx (fun _ : E => p)
(G.measurableSet_infiniteEvent v) (ENNReal.ofReal_pos.mpr hδ)
have hreal : (G.law p).real (H ∆ G.infiniteEvent v) < b*r/(1+r) := by
have hh := (ENNReal.toReal_lt_toReal (measure_ne_top _ _) ENNReal.ofReal_ne_top).mpr happrox
simpa only [law_const,ENNReal.toReal_ofReal hδ.le,Measure.real,law] using hh
filter_upwards [hX S] with n hn
have hh := G.smallFreshBoundary_approx_bound hl p (X n) v M H
(blockSigma_mono hn _ hH)
change r * (G.law p).real (G.infiniteEvent v ∩ G.smallFreshBoundary (X n) v M) ≤
(1+r) * (G.law p).real (H ∆ G.infiniteEvent v) at hh
have hmul := (mul_lt_mul_of_pos_left hreal (show 0<1+r by linarith))
rw [mul_div_cancel₀ _ (ne_of_gt (show 0<1+r by linarith))] at hmul
nlinarith only [hh,hmul,hr]
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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