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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Classification.RawClassificationBoundary
import OAI.Probability.CriticalPercolation.GraphGeometry.GraphQuotientPolynomial
namespace OAI
section
noncomputable section
open Set Function Topology
namespace CriticalPercolation
namespace GraphTopology
variable {X : Type*} (Γ : SimpleGraph X)
def HasWeakGromovQuotientCounterexample : 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)) ∧
Group.FG ((Γ ≃g Γ) ⧸ H) ∧
CompactQuotient.HasSparsePolynomialWordGrowth ((Γ ≃g Γ) ⧸ H) ∧
¬Group.IsVirtuallyNilpotent ((Γ ≃g Γ) ⧸ H)
end GraphTopology
namespace BondGraph
variable {V E : Type*} (G : BondGraph V E)
theorem main_failure_raw_Carolino_or_weak_Gromov [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)) ∧
((¬(let _ : TopologicalSpace (MulAction.orbit G.Aut o) := ⊥
let _ : TopologicalSpace (G.orbitPower o (2*C+1) ≃g G.orbitPower o (2*C+1)) :=
GraphTopology.autTopology (G.orbitPower o (2*C+1))
CompactQuotient.HasUniformCompactLocallyConnectedModels
(G.orbitPower o (2*C+1) ≃g G.orbitPower o (2*C+1)) (5^(d+2)))) ∨
GraphTopology.HasWeakGromovQuotientCounterexample (G.orbitPower o (2*C+1))) := by
obtain ⟨d,hd,C,hconn,hinf,hloc,ht,hpoly,hpow,hfail⟩ :=
G.main_failure_raw_Carolino_or_finite_BGT hc hl hq hpc hper o
refine ⟨d,hd,C,hconn,hinf,hloc,ht,hpoly,hpow,?_⟩
rcases hfail with hcar | hfin
· exact Or.inl hcar
· right
let u : MulAction.orbit G.Aut o := ⟨o,MulAction.mem_orbit_self o⟩
exact GraphTopology.weak_Gromov_counterexample_of_finite_classification_failure
_ hconn hloc ht u (d+1) (fun R => by
obtain ⟨n,hn,hp⟩ := hpoly R
exact ⟨n,hn,hp u⟩) hfin
end BondGraph
end CriticalPercolation
end
end
end OAI
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