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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 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 | import Mathlib
import OAI.Probability.CriticalPercolation.Model
import OAI.Probability.CriticalPercolation.Classification.FaithfulOrbitalModel
import OAI.Probability.CriticalPercolation.Criticality.MainSparseBoundary
import OAI.Probability.CriticalPercolation.GraphGeometry.FullOrbitalStructureTransfer
import OAI.Probability.CriticalPercolation.GraphTopology.GraphGeneratingNeighborhood
namespace OAI
section
noncomputable section
open Set Function MeasureTheory Topology
open scoped Classical ENNReal Pointwise
namespace CriticalPercolation
namespace GraphTopology
variable {X : Type*} (Γ : SimpleGraph X)
def HasCompactOpenVirtuallyNilpotentQuotient : Prop :=
let _ : TopologicalSpace X := ⊥
let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ
∃ H : Subgroup (Γ ≃g Γ), ∃ hn : H.Normal,
let _ := hn
IsCompact (H : Set (Γ ≃g Γ)) ∧ IsOpen (H : Set (Γ ≃g Γ)) ∧
∃ N : Subgroup ((Γ ≃g Γ) ⧸ H), Group.IsNilpotent N ∧ N.index≠0
def HasUniformCompactApproximatePowers (K : ℝ) : Prop :=
let _ : TopologicalSpace X := ⊥
let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ
∃ S : Set (Γ ≃g Γ), IsCompact S ∧ IsOpen S ∧ (1 : Γ ≃g Γ)∈S ∧
S⁻¹=S ∧ Subgroup.closure S=⊤ ∧
∀ R : ℕ, ∃ n≥R, IsCompact (S^n) ∧ IsOpen (S^n) ∧
IsApproximateSubgroup K (S^n)
lemma uniform_compact_approximate_powers_of_sparse_polynomial
(hc : Γ.Connected) (hl : ∀ x, (Γ.neighborSet x).Finite)
(ht : ∀ u v, ∃ g : Γ ≃g Γ, g u=v) (o : X) (d : ℕ)
(hpoly : ∀ R : ℕ, ∃ n≥R, (reachBall Γ o n).ncard≤n^d) :
HasUniformCompactApproximatePowers Γ (5^(d+1)) := by
let _ : TopologicalSpace X := ⊥
let _ : DiscreteTopology X := ⟨rfl⟩
let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ
exact ⟨displacement Γ o 1,
compact_open_generating_approximate_powers Γ hc hl ht o d hpoly⟩
end GraphTopology
namespace BondGraph
variable {V E : Type*} (G : BondGraph V E)
theorem main_failure_topological_classification_counterexample [Infinite V]
(hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive)
(hpc : G.criticalProbability<1)
(hper : G.law G.criticalProbability G.percolates≠0) (o : V) :
∃ d : ℕ, 2≤d ∧ ∃ C : ℕ,
(G.orbitPower o (2*C+1)).Connected ∧
(MulAction.orbit G.Aut o).Infinite ∧
(∀ u, ((G.orbitPower o (2*C+1)).neighborSet u).Finite) ∧
(∀ u v, ∃ g : G.orbitPower o (2*C+1) ≃g G.orbitPower o (2*C+1), g u=v) ∧
(∀ R : ℕ, ∃ n≥R, ∀ u : MulAction.orbit G.Aut o,
(GraphTopology.reachBall (G.orbitPower o (2*C+1)) u n).ncard≤n^(d+1)) ∧
GraphTopology.HasUniformCompactApproximatePowers
(G.orbitPower o (2*C+1)) (5^(d+2)) ∧
¬GraphTopology.HasCompactOpenVirtuallyNilpotentQuotient
(G.orbitPower o (2*C+1)) := by
obtain ⟨hg,_,d,hd,hpoly⟩ := G.main_failure_sparse_polynomial hc hl hq hpc hper
obtain ⟨C,Λ,hfg,hconn,hinf,hloc,htrans,hker,hreps,hscales⟩ :=
G.sparse_polynomial_faithful_orbital_model hc hl hq o d hpoly
let _ : Group.FG Λ := hfg
have ht : ∀ u v, ∃ g : G.orbitPower o (2*C+1) ≃g G.orbitPower o (2*C+1), g u=v := by
intro u v
obtain ⟨g,hg⟩ := htrans u v
exact ⟨g.val,hg⟩
have hpow : GraphTopology.HasUniformCompactApproximatePowers
(G.orbitPower o (2*C+1)) (5^(d+2)) := by
apply GraphTopology.uniform_compact_approximate_powers_of_sparse_polynomial
_ hconn hloc ht (⟨o,MulAction.mem_orbit_self o⟩ : MulAction.orbit G.Aut o) (d+1)
intro R
obtain ⟨n,hn,hh⟩ := hscales R
exact ⟨n,hn,hh _⟩
refine ⟨d,hd,C,hconn,hinf,hloc,ht,hscales,hpow,?_⟩
rintro ⟨H,hn,hcomp,hopen,N,hnil,hN⟩
let _ := hn
let _ := hnil
obtain ⟨S,hS,hcover⟩ := hreps
exact hper (G.critical_null_of_compact_open_orbital_quotient hc hl hg o (2*C+1)
hconn Λ S hS hcover H N hN ⟨hcomp,hopen⟩ hpc)
end BondGraph
end CriticalPercolation
end
end
end OAI
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