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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Model
import OAI.Probability.CriticalPercolation.Criticality.MainResidualBoundary
import OAI.Probability.CriticalPercolation.Modular.CriticalHeavyUniqueness
namespace OAI
section
noncomputable section
open Set MeasureTheory Filter
open scoped Classical ENNReal
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
lemma critical_twoHeavyDistinct_null (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(hq : G.QuasiTransitive) (hp1 : G.criticalProbability<1) (o x y : V) :
G.law G.criticalProbability (G.twoHeavyDistinctEvent o x y)=0 := by
have h := G.atMostOneHeavy_critical_ae hc hl hq hp1 o
have hh : ∀ᵐ ω ∂G.law G.criticalProbability, ω∉G.twoHeavyDistinctEvent o x y := by
filter_upwards [h] with ω hω ⟨⟨hx,hy⟩,hnot⟩
exact hnot (hω x y hx hy)
rw [ae_iff] at hh
simpa only [not_not,Set.ofPred_mem_eq] using hh
theorem main_light_or_sparse_polynomial_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 ∧
∃ v, 0<G.law G.criticalProbability (G.lightInfiniteEvent o v)) ∨
(G.SubexponentialGrowth ∧ ∃ d : ℕ, 2≤d ∧
∀ R : ℕ, ∃ n≥R, ∃ v : V, (G.vertexBall v n).ncard≤n^d) := by
rcases G.main_residual_boundary hc hl hq hpc o hper with ⟨hu,hlight|hheavy⟩|hsparse
· exact Or.inl ⟨hu,hlight⟩
· obtain ⟨x,y,hxy⟩ := hheavy
rw [G.critical_twoHeavyDistinct_null hc hl hq hpc o x y] at hxy
exact False.elim ((lt_irrefl _) hxy)
· exact Or.inr hsparse
end CriticalPercolation.BondGraph
end
end
end OAI
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