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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Criticality.SubcriticalLocalRateBound
import OAI.Probability.CriticalPercolation.Clusters.ClusterNumber
namespace OAI
section
noncomputable section
open Set MeasureTheory
open scoped Classical
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
lemma measurable_infinitelyManyInfiniteClusters [Countable V] [Countable E] :
Measurable (fun ω => G.infinitelyManyInfiniteClusters ω) := by
simp_rw [G.infinitelyManyInfiniteClusters_iff_no_finite_cover]
exact .forall fun S => (G.measurable_infiniteClusterCover (S : Set V)).not
lemma infinitelyManyInfiniteClusters.exists_pairwise {ω : Set E}
(h : G.infinitelyManyInfiniteClusters ω) (n : ℕ) :
∃ f : Fin n → V, (∀ i, (G.cluster ω (f i)).Infinite) ∧
Pairwise (fun i j => ¬ (G.openGraph ω).Reachable (f i) (f j)) := by
induction n with
| zero => exact ⟨Fin.elim0,fun i => Fin.elim0 i,fun i => Fin.elim0 i⟩
| succ n ih =>
obtain ⟨f,hf,hpair⟩ := ih
obtain ⟨v,hv,hnew⟩ := h (Finset.univ.image f)
refine ⟨Fin.cons v f,?_,?_⟩
· intro i
refine Fin.cases hv (fun j => ?_) i
exact hf j
· intro i j hij
revert hij
refine Fin.cases ?_ (fun a => ?_) i
· refine Fin.cases ?_ (fun b hij => ?_) j
· exact fun hij => False.elim (hij rfl)
· exact hnew (f b) (Finset.mem_image.mpr ⟨b,Finset.mem_univ _,rfl⟩)
· refine Fin.cases ?_ (fun b hij => ?_) j
· exact fun _ hab => hnew (f a) (Finset.mem_image.mpr ⟨a,Finset.mem_univ _,rfl⟩) hab.symm
· exact hpair (fun he => hij (congrArg Fin.succ he))
theorem exponential_critical_infinitelyMany_ae [Infinite V]
(hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive)
(o : V) (γ : ℝ) (hγ : 0 < γ) (N : ℕ)
(hgrowth : ∀ m≥N, ∀ v : V, γ*m ≤ Real.log ((G.vertexBall v m).ncard : ℝ))
(hper : G.law G.criticalProbability G.percolates ≠ 0) :
∀ᵐ ω ∂G.law G.criticalProbability, G.infinitelyManyInfiniteClusters ω := by
let := G.countable_vertices hc hl
let := G.countable_bonds hc hl
have hp : (0 : ℝ) < G.criticalProbability := by
have hpc : 0 < G.criticalProbability := by
by_contra h
have he : G.criticalProbability = 0 := le_antisymm (not_lt.mp h) bot_le
exact hper (he ▸ G.law_zero_percolates)
exact hpc
rcases G.unique_or_infinitely_many_ae hc hl hq G.criticalProbability hp with hu | hm
· exact False.elim (hper (Sharpness.critical_null_of_unique G hc hl hq o γ hγ N hgrowth hu))
· exact hm
end CriticalPercolation.BondGraph
end
end
end OAI
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