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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Criticality.FavoredFinite
import OAI.Probability.CriticalPercolation.GraphGeometry.EllipseExit
import OAI.Probability.CriticalPercolation.GraphGeometry.GraphBalls
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory ProbabilityTheory Filter
open scoped Classical Topology ENNReal
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
def seedInfiniteEvent (A : Set V) : Set (Set E) :=
{ω | ∃ a∈A, (G.cluster ω a).Infinite}
lemma seedInfiniteEvent_measurable [Countable V] [Countable E] (A : Set V) :
MeasurableSet (G.seedInfiniteEvent A) := by
have he : G.seedInfiniteEvent A=⋃ a∈A, {ω : Set E | (G.cluster ω a).Infinite} := by ext ω; simp [seedInfiniteEvent]
rw [he]
exact MeasurableSet.iUnion fun a => MeasurableSet.iUnion fun _ => G.measurableSet_infiniteEvent a
lemma seedInfiniteEvent_mono {A B : Set V} (hAB : A⊆B) :
G.seedInfiniteEvent A⊆G.seedInfiniteEvent B := by
rintro ω ⟨a,ha,hi⟩
exact ⟨a,hAB ha,hi⟩
theorem exists_seed_infinite_probability [Countable V] [Countable E]
(hc : G.fullGraph.Connected) (o : V) (q : E → unitInterval)
(hone : BernoulliProduct.law q G.percolates=1) (eps : ℝ) (heps : 0<eps) :
∃ m : ℕ, 1-eps<(BernoulliProduct.law q).real (G.seedInfiniteEvent (G.vertexBall o m)) := by
let A : ℕ → Set (Set E) := fun n => G.seedInfiniteEvent (G.vertexBall o n)
have hmono : Monotone A := fun _ _ h => G.seedInfiniteEvent_mono (G.vertexBall_mono o h)
have heq : (⋃ n, A n)=G.percolates := by
ext ω
constructor
· intro h
obtain ⟨n,a,_ha,hi⟩ := Set.mem_iUnion.mp h
exact ⟨a,hi⟩
· rintro ⟨a,hi⟩
obtain ⟨n,hn⟩ := G.reachable_mem_vertexBall (hc o a)
exact Set.mem_iUnion.mpr ⟨n,a,hn,hi⟩
have ht := tendsto_measure_iUnion_atTop (μ := BernoulliProduct.law q) hmono
rw [heq,hone] at ht
have hreal := (ENNReal.tendsto_toReal (by simp : (1 : ℝ≥0∞)≠⊤)).comp ht
have hnear : ∀ᶠ n in atTop, 1-eps<(BernoulliProduct.law q).real (A n) :=
hreal.eventually (lt_mem_nhds (by simpa using heps : 1-eps<ENNReal.toReal 1))
exact hnear.exists
theorem exists_favored_ellipse [Countable E]
(Psi : V → ℂ) (A : Set V) (zeta D : ℝ) (hD : 0≤D) (hDz : D≤zeta)
(hPsi : ∀ ⦃x y⦄, G.fullGraph.Adj x y → ‖Psi x-Psi y‖≤D)
(hseed : ∀ a∈A, ‖Psi a‖<1) (S : Finset Circle)
(hcover : ∀ c : Circle, ∃ d∈S, ‖(c : ℂ)-(d : ℂ)‖<zeta)
(q : E → unitInterval) (a : ℝ) (ha : 0≤a)
(hi : 1-a^S.card<(BernoulliProduct.law q).real (G.seedInfiniteEvent A))
(hbound : ∀ᵐ ω ∂BernoulliProduct.law q, ∀ x∈A,
(G.cluster (ω∩G.inducedEdges {x | ‖Psi x‖≤1}) x).Finite) :
∃ c∈S, 1-a<(BernoulliProduct.law q).real (G.ellipseExit Psi A zeta c) := by
have hsub : G.seedInfiniteEvent A ≤ᵐ[BernoulliProduct.law q] ⋃ c∈S, G.ellipseExit Psi A zeta c := by
filter_upwards [hbound] with ω hω
exact fun hi => G.infinite_cluster_ellipseExit Psi A zeta D hD hDz hPsi hseed S hcover hi hω
have hprob := ENNReal.toReal_mono (measure_ne_top _ _) (measure_mono_ae hsub)
exact BernoulliProduct.exists_favored_finite q (fun c => G.ellipseExit Psi A zeta c) S
(fun c _ => G.ellipseExit_upper Psi A zeta c) (fun c _ => G.ellipseExit_finiteWitness Psi A zeta c)
a ha (hi.trans_le hprob)
end CriticalPercolation.BondGraph
end
end
end
end OAI
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