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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 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphGeometry.ComponentAction
import OAI.Probability.CriticalPercolation.Modular.Unimodularity
import OAI.Probability.CriticalPercolation.Modular.ModularCharacter
import OAI.Probability.CriticalPercolation.Modular.ActionUnimodularRoot
namespace OAI
section
noncomputable section
open Set
open scoped Classical
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
@[instance_reducible] def autEdgeAction : MulAction G.Aut E where
smul φ e := φ.edge e
one_smul _ := rfl
mul_smul _ _ _ := rfl
local instance : MulAction G.Aut E := G.autEdgeAction
lemma aut_preservesEnds : G.PreservesEnds (Λ := G.Aut) := fun φ => φ.preserves
lemma stabilizer_le_levelComponentGroup (hc : G.fullGraph.Connected)
(hl : G.LocallyFinite) (o : V) (Q : Set ℝˣ) (v : V)
(x : G.domainComponent (G.modularCoordinate hc hl o ⁻¹' Q) v) :
MulAction.stabilizer G.Aut x.val ≤ componentKernel (G.modularUnitCharacter hc hl o)
(G.domainComponent (G.modularCoordinate hc hl o ⁻¹' Q) v) := by
intro φ hφ
have hker := G.unit_stabilizer_le_kernel hc hl o x.val hφ
change φ.vertex x.val = x.val at hφ
refine ⟨hker,?_⟩
have hS : ∀ y, φ.vertex y ∈ G.modularCoordinate hc hl o ⁻¹' Q ↔
y ∈ G.modularCoordinate hc hl o ⁻¹' Q := by
intro y
change G.modularCoordinate hc hl o (φ • y) ∈ Q ↔ _
rw [G.modularCoordinate_equivariant hc hl o φ y,hker,one_mul]
rfl
exact Aut.preserves_domainComponent G φ _ hS v x.val x.property
(by rw [hφ]; exact x.property)
lemma levelComponent_suborbit_image (hc : G.fullGraph.Connected)
(hl : G.LocallyFinite) (o : V) (Q : Set ℝˣ) (v : V) :
let C := G.domainComponent (G.modularCoordinate hc hl o ⁻¹' Q) v
let L := componentKernel (G.modularUnitCharacter hc hl o) C
let _ := restrictedVertexAction C L (componentKernel_le _ _)
∀ x y : C, Subtype.val '' ActionOrbitTransport.stabilizerOrbit L C x y =
G.stabilizerOrbit x.val y.val := by
intro C L _ x y
ext z
constructor
· rintro ⟨w,hw,rfl⟩
obtain ⟨φ,hx,hy⟩ := (ActionOrbitTransport.mem_stabilizerOrbit L C).mp hw
exact G.mem_stabilizerOrbit.mpr ⟨φ.val,congrArg Subtype.val hx,congrArg Subtype.val hy⟩
· intro hz
obtain ⟨φ,hx,hy⟩ := G.mem_stabilizerOrbit.mp hz
have hm : φ ∈ L := G.stabilizer_le_levelComponentGroup hc hl o Q v x hx
let ψ : L := ⟨φ,hm⟩
refine ⟨ψ • y,?_,hy⟩
exact (ActionOrbitTransport.mem_stabilizerOrbit L C).mpr
⟨ψ,Subtype.ext hx,rfl⟩
lemma levelComponent_suborbit_finite (hc : G.fullGraph.Connected)
(hl : G.LocallyFinite) (o : V) (Q : Set ℝˣ) (v : V) :
let C := G.domainComponent (G.modularCoordinate hc hl o ⁻¹' Q) v
let L := componentKernel (G.modularUnitCharacter hc hl o) C
let _ := restrictedVertexAction C L (componentKernel_le _ _)
∀ x y : C, (ActionOrbitTransport.stabilizerOrbit L C x y).Finite := by
intro C L _ x y
apply Set.Finite.of_finite_image (f := Subtype.val) ?_ Subtype.val_injective.injOn
rw [G.levelComponent_suborbit_image hc hl o Q v]
exact G.stabilizerOrbit_finite hc hl x.val y.val
lemma levelComponent_suborbitCard (hc : G.fullGraph.Connected)
(hl : G.LocallyFinite) (o : V) (Q : Set ℝˣ) (v : V) :
let C := G.domainComponent (G.modularCoordinate hc hl o ⁻¹' Q) v
let L := componentKernel (G.modularUnitCharacter hc hl o) C
let _ := restrictedVertexAction C L (componentKernel_le _ _)
∀ x y : C, ActionOrbitTransport.suborbitCard L C x y = G.suborbitCard x.val y.val := by
intro C L _ x y
unfold ActionOrbitTransport.suborbitCard
rw [←Set.ncard_image_of_injective _ Subtype.val_injective,
G.levelComponent_suborbit_image hc hl o Q v]
rfl
def levelComponent_unimodularRoot (hc : G.fullGraph.Connected)
(hl : G.LocallyFinite) (hq : G.QuasiTransitive) (o : V)
(Q : Set ℝˣ) (hQ : Q.Finite) (v : V) (hv : G.modularCoordinate hc hl o v ∈ Q) :
(G.induce (G.domainComponent (G.modularCoordinate hc hl o ⁻¹' Q) v)).UnimodularRoot := Classical.choice (by
let C := G.domainComponent (G.modularCoordinate hc hl o ⁻¹' Q) v
let L := componentKernel (G.modularUnitCharacter hc hl o) C
let := restrictedVertexAction C L (componentKernel_le _ _)
let := G.restrictedEdgeAction G.aut_preservesEnds C L (componentKernel_le _ _)
have ha := G.restricted_preservesEnds G.aut_preservesEnds C L (componentKernel_le _ _)
obtain ⟨R,hR,hcover⟩ := hq
have hcover' : ∀ x, ∃ u∈R, x∈MulAction.orbit G.Aut u := by
intro x
obtain ⟨u,hu,φ,hφ⟩ := hcover x
exact ⟨u,hu,φ,hφ⟩
obtain ⟨S,hS,hScover⟩ := G.componentKernel_finite_orbits G.aut_preservesEnds
R hR hcover' (G.modularUnitCharacter hc hl o) (G.modularCoordinate hc hl o)
(G.modularCoordinate_equivariant hc hl o) Q hQ v hv
let w : C → ℝ := fun x => G.modularRatio o x.val
have hw (φ : L) (x : C) : w (φ • x) = w x := by
have hchar : G.modularUnitCharacter hc hl o φ.val = 1 := φ.property.1
have hh := congrArg (fun u : ℝˣ => (u : ℝ))
(G.modularCoordinate_equivariant hc hl o φ.val x.val)
change G.modularRatio o (φ.val.vertex x.val) = _ at hh
change G.modularRatio o (φ.val.vertex x.val) = _
simpa only [hchar,one_mul,modularCoordinate_val] using hh
have hb (x y : C) : w x * (ActionOrbitTransport.suborbitCard L C x y : ℝ) =
w y * (ActionOrbitTransport.suborbitCard L C y x : ℝ) := by
rw [G.levelComponent_suborbitCard hc hl o Q v,
G.levelComponent_suborbitCard hc hl o Q v]
exact G.modularRatio_card_balance hc hl o x.val y.val
exact ⟨unimodularRoot_of_action (G.induce C) ha S hS hScover
⟨v,G.domainComponent_self _ _⟩ w (fun x => G.modularRatio_pos hc hl o x.val)
hw (G.levelComponent_suborbit_finite hc hl o Q v) hb⟩)
end CriticalPercolation.BondGraph
end
end
end OAI
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