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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 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphTopology.GraphNeighborhoodCompact
import OAI.Probability.CriticalPercolation.GraphGeometry.GraphBalls
import OAI.Probability.CriticalPercolation.GraphGeometry.OrbitalAction
import OAI.Probability.CriticalPercolation.GraphGeometry.OrbitQuotientTransfer
import OAI.Probability.CriticalPercolation.Modular.ModularLevelGeometry
import OAI.Probability.CriticalPercolation.Nilpotent.StructuredVirtuallyNilpotent
namespace OAI
section
noncomputable section
open Set Function MeasureTheory Topology
open scoped Classical ENNReal
namespace CriticalPercolation
namespace StructurePullback
variable {A B : Type*} [Group A] [Group B]
lemma nilpotent_comap_of_injective (f : A →* B) (hf : Injective f)
(N : Subgroup B) [Group.IsNilpotent N] : Group.IsNilpotent (N.comap f) := by
let F : N.comap f →* N :=
{toFun := fun g => ⟨f g.val,g.property⟩
map_one' := Subtype.ext (map_one f)
map_mul' := fun g h => Subtype.ext (map_mul f g.val h.val)}
have hF : Injective F := by
intro g h he
apply Subtype.ext
exact hf (congrArg Subtype.val he)
apply Subgroup.isNilpotent_of_ker_le_center F
rw [(MonoidHom.ker_eq_bot_iff F).mpr hF]
exact bot_le
lemma finiteIndex_comap (f : A →* B) (N : Subgroup B) (hN : N.index≠0) :
(N.comap f).index≠0 := by
rw [Subgroup.index_comap]
intro h
exact hN (Subgroup.index_eq_zero_of_relIndex_eq_zero h)
end StructurePullback
namespace BondGraph
variable {V E : Type*} (G : BondGraph V E)
lemma finite_stabilizer_images_from_one_vertex
{A B : Type*} [Group A] [Group B] [MulAction A V] [MulAction A E]
(hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(ha : G.PreservesEnds (Λ := A)) (ρ : A →* B) (o : V)
(ho : ((MulAction.stabilizer A o).map ρ : Set B).Finite) :
∀ v : V, ((MulAction.stabilizer A v).map ρ : Set B).Finite := by
let a : V → A := fun w => if h : ∃ g : A, g • o=w then h.choose else 1
have ha' : ∀ {w : V}, (∃ g : A, g • o=w) → a w • o=w := by
intro w hw
simp only [a, dite_eq_left hw]
exact hw.choose_spec
intro v
obtain ⟨n,hn⟩ := G.reachable_mem_vertexBall (hc v o)
apply ((G.vertexBall_finite hl v n).biUnion
(fun w _ => ho.image (fun b => ρ (a w)*b))).subset
rintro b ⟨g,hg,rfl⟩
have hgo : g • o∈G.vertexBall v n := by
have hh := (Aut.vertexBall_mem G (G.smulAut ha g) v o n).mpr hn
change g • o∈G.vertexBall (g • v) n at hh
exact hg ▸ hh
refine Set.mem_iUnion₂.mpr ⟨g • o,hgo,ρ ((a (g • o))⁻¹*g),?_,?_⟩
· refine ⟨(a (g • o))⁻¹*g,?_,rfl⟩
change ((a (g • o))⁻¹*g) • o=o
rw [mul_smul]
calc
(a (g • o))⁻¹ • (g • o) = (a (g • o))⁻¹ • (a (g • o) • o) :=
congrArg (fun x => (a (g • o))⁻¹ • x) (ha' ⟨g,rfl⟩).symm
_ = o := inv_smul_smul _ _
· simp only [map_mul, map_inv, mul_inv_cancel_left]
def fullOrbitalQuotientHom (o : V) (L : ℕ) (Λ : Subgroup G.Aut)
(H : Subgroup (G.orbitPower o L ≃g G.orbitPower o L)) [H.Normal] :
Λ →* (G.orbitPower o L ≃g G.orbitPower o L) ⧸ H :=
(QuotientGroup.mk' H).comp ((G.orbitPowerHom o L).comp Λ.subtype)
lemma fullOrbital_finite_kernel_orbits (hc : G.fullGraph.Connected)
(hl : G.LocallyFinite) (o : V) (L : ℕ) (Λ : Subgroup G.Aut)
(H : Subgroup (G.orbitPower o L ≃g G.orbitPower o L)) [H.Normal]
(hH : ((fun g : G.orbitPower o L ≃g G.orbitPower o L =>
g (⟨o,MulAction.mem_orbit_self o⟩ : MulAction.orbit G.Aut o)) '' (H : Set _)).Finite) :
∀ v : V, (MulAction.orbit (G.fullOrbitalQuotientHom o L Λ H).rangeRestrict.ker v).Finite := by
let _ : MulAction G.Aut E := G.autEdgeAction
apply G.finite_kernel_orbits_from_one_vertex hc hl
(G.subgroup_preservesEnds G.aut_preservesEnds Λ) _ o
apply (hH.image Subtype.val).subset
rintro v ⟨g,rfl⟩
have he := congrArg Subtype.val g.property
have hg : G.orbitPowerHom o L g.val.val∈H := (QuotientGroup.eq_one_iff _).mp he
exact ⟨⟨g • o,⟨g.val.val,rfl⟩⟩,⟨G.orbitPowerHom o L g.val.val,hg,rfl⟩,rfl⟩
lemma fullOrbital_finite_stabilizer_images (hc : G.fullGraph.Connected)
(hl : G.LocallyFinite) (o : V) (L : ℕ) (Λ : Subgroup G.Aut)
(H : Subgroup (G.orbitPower o L ≃g G.orbitPower o L)) [H.Normal]
(hstab : (((QuotientGroup.mk' H) : (G.orbitPower o L ≃g G.orbitPower o L) → _)
'' {g | g (⟨o,MulAction.mem_orbit_self o⟩ : MulAction.orbit G.Aut o)=
⟨o,MulAction.mem_orbit_self o⟩}).Finite) :
∀ v : V, ((MulAction.stabilizer Λ v).map
(G.fullOrbitalQuotientHom o L Λ H).rangeRestrict :
Set (G.fullOrbitalQuotientHom o L Λ H).range).Finite := by
let _ : MulAction G.Aut E := G.autEdgeAction
apply G.finite_stabilizer_images_from_one_vertex hc hl
(G.subgroup_preservesEnds G.aut_preservesEnds Λ) _ o
apply (hstab.preimage Subtype.val_injective.injOn).subset
rintro g ⟨a,ha,rfl⟩
refine ⟨G.orbitPowerHom o L a.val,?_,rfl⟩
apply Subtype.ext
exact ha
theorem critical_null_of_compact_open_orbital_quotient
(hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hg : G.SubexponentialGrowth)
(o : V) (L : ℕ) (hconn : (G.orbitPower o L).Connected)
(Λ : Subgroup G.Aut) [Group.FG Λ]
(S : Set V) (hS : S.Finite) (hcover : ∀ v, ∃ s∈S, v∈MulAction.orbit Λ s)
(H : Subgroup (G.orbitPower o L ≃g G.orbitPower o L)) [H.Normal]
(N : Subgroup ((G.orbitPower o L ≃g G.orbitPower o L) ⧸ H)) [Group.IsNilpotent N]
(hN : N.index≠0)
(hH : let _ : TopologicalSpace (MulAction.orbit G.Aut o) := ⊥
let _ : TopologicalSpace (G.orbitPower o L ≃g G.orbitPower o L) :=
GraphTopology.autTopology (G.orbitPower o L)
IsCompact (H : Set (G.orbitPower o L ≃g G.orbitPower o L)) ∧
IsOpen (H : Set (G.orbitPower o L ≃g G.orbitPower o L)))
(hpc : G.criticalProbability<1) : G.law G.criticalProbability G.percolates=0 := by
let _ : TopologicalSpace (MulAction.orbit G.Aut o) := ⊥
let _ : DiscreteTopology (MulAction.orbit G.Aut o) := ⟨rfl⟩
let _ : TopologicalSpace (G.orbitPower o L ≃g G.orbitPower o L) :=
GraphTopology.autTopology (G.orbitPower o L)
let _ : DiscreteTopology ((G.orbitPower o L ≃g G.orbitPower o L) ⧸ H) :=
QuotientGroup.discreteTopology hH.2
let o' : MulAction.orbit G.Aut o := ⟨o,MulAction.mem_orbit_self o⟩
have hstab : (((QuotientGroup.mk' H) : (G.orbitPower o L ≃g G.orbitPower o L) → _)
'' {g | g o'=o'}).Finite := by
apply isCompact_iff_finite.mp
apply (GraphTopology.compact_stabilizer (G.orbitPower o L) hconn
(G.powerOn_finite_neighbors hl _ _) o').image
exact QuotientGroup.continuous_mk
let f := G.fullOrbitalQuotientHom o L Λ H
let M := N.comap f.range.subtype
let _ : Group.IsNilpotent M := StructurePullback.nilpotent_comap_of_injective
f.range.subtype Subtype.val_injective N
let _ : MulAction G.Aut E := G.autEdgeAction
exact G.critical_null_of_virtually_nilpotent_quotient hc hl hg
(G.subgroup_preservesEnds G.aut_preservesEnds Λ) S hS hcover
f.rangeRestrict (MonoidHom.rangeRestrict_surjective f)
(G.fullOrbital_finite_kernel_orbits hc hl o L Λ H
(GraphTopology.compact_finite_vertex_images _ _ hH.1 o'))
(G.fullOrbital_finite_stabilizer_images hc hl o L Λ H hstab) M
(StructurePullback.finiteIndex_comap f.range.subtype N hN) hpc
end BondGraph
end CriticalPercolation
end
end
end OAI
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