File size: 4,875 Bytes
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 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 | import Mathlib
import OAI.Probability.CriticalPercolation.Model
import OAI.Probability.CriticalPercolation.GraphGeometry.EquivariantCoordinates
import OAI.Probability.CriticalPercolation.GraphGeometry.GraphActionCoordinates
import OAI.Probability.CriticalPercolation.GraphGeometry.ActionCoordinates
import OAI.Probability.CriticalPercolation.GraphGeometry.CriticalCoordinates
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory ProbabilityTheory
open scoped Classical ENNReal
namespace CriticalPercolation.BondGraph
variable {V E Λ N : Type*} (G : BondGraph V E)
[Group Λ] [Group N] [MulAction Λ V] [MulAction Λ E]
def integerPairComplex (z : ℤ×ℤ) : ℂ := (z.1 : ℂ)+(z.2 : ℂ)*Complex.I
lemma integerPairComplex_add (a b : ℤ×ℤ) :
integerPairComplex (a+b)=integerPairComplex a+integerPairComplex b := by
simp only [integerPairComplex,Prod.fst_add,Prod.snd_add,Int.cast_add]
ring
lemma integerPairComplex_re (z : ℤ×ℤ) : (integerPairComplex z).re=z.1 := by
simp [integerPairComplex]
lemma integerPairComplex_im (z : ℤ×ℤ) : (integerPairComplex z).im=z.2 := by
simp [integerPairComplex]
theorem exists_integer_pair_coordinates
(ρ : Λ →* N) (hρ : Surjective ρ)
(φ : N →* Multiplicative (ℤ×ℤ)) (hφ : Surjective φ)
(hstab : ∀ v : V, ((MulAction.stabilizer Λ v).map ρ : Set N).Finite)
[Nonempty V] :
∃ z : V → ℤ×ℤ, Surjective z ∧
∀ g v, z (g • v)=(φ (ρ g)).toAdd+z v := by
obtain ⟨π,hπ⟩ := EquivariantCoordinates.exists_equivariant (φ.comp ρ)
(EquivariantCoordinates.stabilizer_killed ρ φ hstab)
refine ⟨fun v => (π v).toAdd,?_,?_⟩
· intro t
obtain ⟨v,hv⟩ := EquivariantCoordinates.surjective_of_equivariant
(φ.comp ρ) (hφ.comp hρ) π hπ (Multiplicative.ofAdd t)
exact ⟨v,congrArg Multiplicative.toAdd hv⟩
· intro g v
exact congrArg Multiplicative.toAdd (hπ g v)
lemma integer_pair_coordinate_geometry
(hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(ha : G.PreservesEnds (Λ := Λ))
(S : Set V) (hS : S.Finite) (hcover : ∀ v, ∃ s∈S, v∈MulAction.orbit Λ s)
(a : Λ → ℤ×ℤ) (z : V → ℤ×ℤ) (honto : Surjective z)
(hz : ∀ g v, z (g • v)=a g+z v) :
∃ (o : V) (c : ℕ) (D : ℝ), 0≤D ∧ integerPairComplex (z o)=0 ∧
(∀ ⦃x y⦄, G.fullGraph.Adj x y →
‖integerPairComplex (z x)-integerPairComplex (z y)‖≤D) ∧
(∀ m n : ℤ, ∃ x, (integerPairComplex (z x)).re=m ∧
(integerPairComplex (z x)).im=n) ∧
(∀ y, ∃ psi : G.Aut, psi.vertex o∈G.vertexBall y c ∧
∀ x y, integerPairComplex (z (psi.vertex x))-integerPairComplex (z (psi.vertex y))=
integerPairComplex (z x)-integerPairComplex (z y)) := by
let pi : V → ℂ := fun v => integerPairComplex (z v)
have hpi : ∀ g v, pi (g • v)=integerPairComplex (a g)+pi v := by
intro g v
exact (congrArg integerPairComplex (hz g v)).trans (integerPairComplex_add _ _)
obtain ⟨o,ho⟩ := honto 0
obtain ⟨c,hc'⟩ := G.action_orbit_dense (fun g => G.smulAut ha g) (fun _ _ => rfl)
hc S hS hcover o
obtain ⟨D,hD,hj⟩ := G.bounded_coordinate_jumps hl ha S hS hcover
(fun g => integerPairComplex (a g)) pi hpi
refine ⟨o,c,D,hD,?_,?_,?_,?_⟩
· simp [ho,integerPairComplex]
· intro x y hxy
obtain ⟨⟨e,_,he⟩,_⟩ := (SimpleGraph.fromEdgeSet_adj _).mp hxy
exact hj e x y he
· intro m n
obtain ⟨x,hx⟩ := honto (m,n)
exact ⟨x,by rw [hx,integerPairComplex_re],by rw [hx,integerPairComplex_im]⟩
· intro y
obtain ⟨g,hg⟩ := hc' y
refine ⟨G.smulAut ha g,hg,?_⟩
intro x y
change pi (g • x)-pi (g • y)=pi x-pi y
rw [hpi,hpi]
abel
theorem critical_null_of_action_coordinates
(hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(ha : G.PreservesEnds (Λ := Λ))
(S : Set V) (hS : S.Finite) (hcover : ∀ v, ∃ s∈S, v∈MulAction.orbit Λ s)
(hg : G.SubexponentialGrowth) (hpc : G.criticalProbability<1)
(a : Λ → ℤ×ℤ) (z : V → ℤ×ℤ) (honto : Surjective z)
(hz : ∀ g v, z (g • v)=a g+z v)
(hnoray : ∀ᵐ ω ∂G.law G.criticalProbability,
¬∃ r : G.OpenRay ω, ∃ U W : ℝ, ∀ n,
|(z (r.vertex n)).1|≤U ∧ |(z (r.vertex n)).2|≤W) :
G.law G.criticalProbability G.percolates=0 := by
obtain ⟨o,c,D,hD,ho,hjump,honto',hdense⟩ :=
G.integer_pair_coordinate_geometry hc hl ha S hS hcover a z honto hz
apply G.critical_null_of_unbounded_projection hc hl
(G.quasiTransitive_of_action ha S hS hcover) hg hpc
(fun v => integerPairComplex (z v)) o c D hD ho hjump honto' hdense
simpa only [integerPairComplex_re,integerPairComplex_im,Int.cast_abs] using hnoray
end CriticalPercolation.BondGraph
end
end
end
end OAI
|