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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Model
import OAI.Probability.CriticalPercolation.Modular.CriticalHeavyUniqueness
import OAI.Probability.CriticalPercolation.Modular.LightClusterNumber
import OAI.Probability.CriticalPercolation.Modular.MainLightBoundary
namespace OAI
section
noncomputable section
open Set MeasureTheory
open scoped Classical ENNReal
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
theorem main_infinite_light_or_sparse_boundary [Infinite V]
(hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive)
(hpc : G.criticalProbability<1) (o : V)
(hper : G.law G.criticalProbability G.percolates≠0) :
(¬G.CombinatoriallyUnimodular ∧
(∀ᵐ ω ∂G.law G.criticalProbability, G.infinitelyManyLightClusters ω) ∧
(∀ᵐ ω ∂G.law G.criticalProbability, ω∈G.atMostOneHeavy o)) ∨
(G.SubexponentialGrowth ∧ ∃ d : ℕ, 2≤d ∧
∀ R : ℕ, ∃ n≥R, ∃ v : V, (G.vertexBall v n).ncard≤n^d) := by
rcases G.main_light_or_sparse_polynomial_boundary hc hl hq hpc o hper with ⟨hu,v,hv⟩|hpoly
· exact Or.inl ⟨hu,G.infinitely_many_light_ae_of_positive hc hl hq G.criticalProbability o v hv,
G.atMostOneHeavy_critical_ae hc hl hq hpc o⟩
· exact Or.inr hpoly
end CriticalPercolation.BondGraph
end
end
end OAI
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