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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Model
import OAI.Probability.CriticalPercolation.Criticality.MainSparseBoundary
import OAI.Probability.CriticalPercolation.Modular.ModularLevelGeometry
import OAI.Probability.CriticalPercolation.Nilpotent.StructuredVirtuallyNilpotent
namespace OAI
section
noncomputable section
open Set MeasureTheory
open scoped Classical ENNReal
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
theorem critical_null_of_normal_nilpotent_model
(hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(hg : G.SubexponentialGrowth)
(Λ : Subgroup G.Aut) [Group.FG Λ]
(S : Set V) (hS : S.Finite) (hcover : ∀ v, ∃ s∈S, v∈MulAction.orbit Λ s)
(K : Subgroup Λ) [K.Normal]
(hker : ∀ v : V, (MulAction.orbit K v).Finite)
(hstab : ∀ v : V, ((MulAction.stabilizer Λ v).map (QuotientGroup.mk' K) :
Set (Λ ⧸ K)).Finite)
(N : Subgroup (Λ ⧸ K)) [Group.IsNilpotent N] (hN : N.index≠0)
(hpc : G.criticalProbability<1) : G.law G.criticalProbability G.percolates=0 := by
let _ : MulAction G.Aut E := G.autEdgeAction
apply G.critical_null_of_virtually_nilpotent_quotient hc hl hg
(G.subgroup_preservesEnds G.aut_preservesEnds Λ) S hS hcover
(QuotientGroup.mk' K) (QuotientGroup.mk'_surjective K) _ hstab N hN hpc
intro v
apply (hker v).subset
rintro w ⟨g,rfl⟩
have hg : g.val∈K := by simpa only [QuotientGroup.ker_mk'] using g.property
exact ⟨⟨g.val,hg⟩,rfl⟩
theorem main_failure_excludes_normal_nilpotent_model [Infinite V]
(hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive)
(hpc : G.criticalProbability<1)
(hper : G.law G.criticalProbability G.percolates≠0)
(Λ : Subgroup G.Aut) [Group.FG Λ]
(S : Set V) (hS : S.Finite) (hcover : ∀ v, ∃ s∈S, v∈MulAction.orbit Λ s)
(K : Subgroup Λ) [K.Normal]
(hker : ∀ v : V, (MulAction.orbit K v).Finite)
(hstab : ∀ v : V, ((MulAction.stabilizer Λ v).map (QuotientGroup.mk' K) :
Set (Λ ⧸ K)).Finite)
(N : Subgroup (Λ ⧸ K)) [Group.IsNilpotent N] (hN : N.index≠0) : False := by
have hg := (G.main_failure_sparse_polynomial hc hl hq hpc hper).1
exact hper (G.critical_null_of_normal_nilpotent_model hc hl hg Λ S hS hcover K hker hstab N hN hpc)
end CriticalPercolation.BondGraph
end
end
end OAI
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