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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Modular.GraphModular
namespace OAI
section
noncomputable section
open Set
open scoped Classical
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
lemma modularRatio_aut_translate (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(o x : V) (φ : G.Aut) :
G.modularRatio o (φ.vertex x) =
G.modularRatio o (φ.vertex o) * G.modularRatio o x := by
rw [←G.modularRatio_cocycle hc hl o (φ.vertex o) (φ.vertex x),Aut.modularRatio]
lemma modularRatio_aut_base (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(o x : V) (φ : G.Aut) :
G.modularRatio x (φ.vertex x) = G.modularRatio o (φ.vertex o) := by
have h1 := G.modularRatio_cocycle hc hl o x (φ.vertex x)
rw [G.modularRatio_aut_translate hc hl o x φ] at h1
exact (mul_left_cancel₀ (ne_of_gt (G.modularRatio_pos hc hl o x)))
(h1.trans (mul_comm _ _))
def modularCharacter (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (o : V) :
G.Aut →* ℝ where
toFun φ := G.modularRatio o (φ.vertex o)
map_one' := G.modularRatio_self o
map_mul' φ ψ := by
change G.modularRatio o (φ.vertex (ψ.vertex o)) = _
exact G.modularRatio_aut_translate hc hl o (ψ.vertex o) φ
@[simp] lemma modularCharacter_apply (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(o : V) (φ : G.Aut) :
G.modularCharacter hc hl o φ = G.modularRatio o (φ.vertex o) := rfl
lemma modularCharacter_pos (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(o : V) (φ : G.Aut) : 0 < G.modularCharacter hc hl o φ :=
G.modularRatio_pos hc hl o (φ.vertex o)
lemma modularCharacter_base (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(o x : V) : G.modularCharacter hc hl o = G.modularCharacter hc hl x := by
ext φ
exact (G.modularRatio_aut_base hc hl o x φ).symm
def modularKernel (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (o : V) :
Subgroup G.Aut := (G.modularCharacter hc hl o).ker
lemma mem_modularKernel (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(o : V) (φ : G.Aut) : φ ∈ G.modularKernel hc hl o ↔
∀ x, G.modularRatio o (φ.vertex x) = G.modularRatio o x := by
change G.modularCharacter hc hl o φ = 1 ↔ _
constructor
· intro h x
rw [G.modularRatio_aut_translate hc hl o x φ,←G.modularCharacter_apply hc hl,h,one_mul]
· intro h
simpa only [modularCharacter_apply,G.modularRatio_self] using h o
lemma stabilizer_le_modularKernel (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(o x : V) : MulAction.stabilizer G.Aut x ≤ G.modularKernel hc hl o := by
intro φ hφ
change G.modularCharacter hc hl o φ = 1
rw [G.modularCharacter_base hc hl o x,modularCharacter_apply]
change φ.vertex x = x at hφ
rw [hφ,G.modularRatio_self]
lemma modularKernel_transitive_on_level (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(o x y : V) (hxy : ∃ φ : G.Aut, φ.vertex x = y)
(hlevel : G.modularRatio o x = G.modularRatio o y) :
∃ φ : G.modularKernel hc hl o, φ.val.vertex x = y := by
obtain ⟨φ,hφ⟩ := hxy
refine ⟨⟨φ,?_⟩,hφ⟩
change G.modularRatio o (φ.vertex o) = 1
have hh := G.modularRatio_aut_translate hc hl o x φ
rw [hφ,←hlevel] at hh
exact (mul_right_cancel₀ (ne_of_gt (G.modularRatio_pos hc hl o x)))
(hh.symm.trans (one_mul _).symm)
def modularCoordinate (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(o x : V) : ℝˣ := Units.mk0 (G.modularRatio o x)
(ne_of_gt (G.modularRatio_pos hc hl o x))
@[simp] lemma modularCoordinate_val (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(o x : V) : (G.modularCoordinate hc hl o x : ℝ) = G.modularRatio o x := rfl
def modularUnitCharacter (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (o : V) :
G.Aut →* ℝˣ where
toFun φ := G.modularCoordinate hc hl o (φ.vertex o)
map_one' := Units.ext (G.modularRatio_self o)
map_mul' φ ψ := Units.ext (G.modularRatio_aut_translate hc hl o (ψ.vertex o) φ)
lemma modularCoordinate_equivariant (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(o : V) (φ : G.Aut) (x : V) :
G.modularCoordinate hc hl o (φ • x) =
G.modularUnitCharacter hc hl o φ * G.modularCoordinate hc hl o x :=
Units.ext (G.modularRatio_aut_translate hc hl o x φ)
lemma unit_stabilizer_le_kernel (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(o x : V) : MulAction.stabilizer G.Aut x ≤ (G.modularUnitCharacter hc hl o).ker := by
intro φ hφ
apply Units.ext
exact G.stabilizer_le_modularKernel hc hl o x hφ
end CriticalPercolation.BondGraph
end
end
end OAI
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