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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 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphGeometry.GraphAction
import OAI.Probability.CriticalPercolation.GraphGeometry.GraphBalls
import OAI.Probability.CriticalPercolation.Modular.RelativeModular
namespace OAI
section
noncomputable section
open Set
open scoped Classical
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
def stabilizerOrbit (x y : V) : Set V :=
MulAction.orbit (MulAction.stabilizer G.Aut x) y
lemma mem_stabilizerOrbit {x y z : V} : z ∈ G.stabilizerOrbit x y ↔
∃ φ : G.Aut, φ.vertex x = x ∧ φ.vertex y = z := by
simp only [stabilizerOrbit, MulAction.mem_orbit_iff]
constructor
· rintro ⟨φ,hφ⟩
exact ⟨φ,φ.property,hφ⟩
· rintro ⟨φ,hx,hy⟩
exact ⟨⟨φ,hx⟩,hy⟩
lemma self_mem_stabilizerOrbit (x y : V) : y ∈ G.stabilizerOrbit x y :=
(G.mem_stabilizerOrbit).mpr ⟨1,rfl,rfl⟩
lemma stabilizerOrbit_finite (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(x y : V) : (G.stabilizerOrbit x y).Finite := by
obtain ⟨n,hn⟩ := G.finite_subset_vertexBall hc x (Set.finite_singleton y)
apply (G.vertexBall_finite hl x n).subset
intro z hz
obtain ⟨φ,hx,rfl⟩ := G.mem_stabilizerOrbit.mp hz
rw [←hx]
exact (Aut.vertexBall_mem G φ x y n).mpr (hn (Set.mem_singleton y))
def suborbitCard (x y : V) : ℕ := (G.stabilizerOrbit x y).ncard
lemma suborbitCard_pos (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(x y : V) : 0 < G.suborbitCard x y :=
(Set.ncard_pos (G.stabilizerOrbit_finite hc hl x y)).mpr ⟨y,G.self_mem_stabilizerOrbit x y⟩
lemma suborbitCard_eq_relIndex (x y : V) : G.suborbitCard x y =
(MulAction.stabilizer G.Aut y).relIndex (MulAction.stabilizer G.Aut x) := by
have heq : (MulAction.stabilizer G.Aut y).subgroupOf (MulAction.stabilizer G.Aut x) =
MulAction.stabilizer (MulAction.stabilizer G.Aut x) y := by
ext φ
rfl
rw [Subgroup.relIndex,heq,MulAction.index_stabilizer]
rfl
lemma Aut.stabilizerOrbit_image (φ : G.Aut) (x y : V) :
φ.vertex '' G.stabilizerOrbit x y = G.stabilizerOrbit (φ.vertex x) (φ.vertex y) := by
ext z
constructor
· rintro ⟨w,hw,rfl⟩
obtain ⟨ψ,hx,hy⟩ := G.mem_stabilizerOrbit.mp hw
apply G.mem_stabilizerOrbit.mpr
refine ⟨φ * ψ * φ⁻¹,?_,?_⟩
· change φ.vertex (ψ.vertex (φ.vertex.symm (φ.vertex x))) = φ.vertex x
rw [Equiv.symm_apply_apply,hx]
· change φ.vertex (ψ.vertex (φ.vertex.symm (φ.vertex y))) = φ.vertex w
rw [Equiv.symm_apply_apply,hy]
· intro hz
obtain ⟨ψ,hx,hy⟩ := G.mem_stabilizerOrbit.mp hz
refine ⟨φ.vertex.symm z,?_,φ.vertex.apply_symm_apply z⟩
apply G.mem_stabilizerOrbit.mpr
refine ⟨φ⁻¹ * ψ * φ,?_,?_⟩
· change φ.vertex.symm (ψ.vertex (φ.vertex x)) = x
rw [hx,Equiv.symm_apply_apply]
· change φ.vertex.symm (ψ.vertex (φ.vertex y)) = φ.vertex.symm z
rw [hy]
lemma Aut.suborbitCard (φ : G.Aut) (x y : V) :
G.suborbitCard (φ.vertex x) (φ.vertex y) = G.suborbitCard x y := by
rw [CriticalPercolation.BondGraph.suborbitCard,←Aut.stabilizerOrbit_image G φ,
Set.ncard_image_of_injective _ φ.vertex.injective]
rfl
def modularRatio (x y : V) : ℝ := (G.suborbitCard x y : ℝ) / (G.suborbitCard y x : ℝ)
lemma modularRatio_eq_relativeMass (x y : V) : G.modularRatio x y =
Subgroup.relativeMass (MulAction.stabilizer G.Aut x) (MulAction.stabilizer G.Aut y) := by
simp only [modularRatio,suborbitCard_eq_relIndex,Subgroup.relativeMass]
lemma modularRatio_pos (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (x y : V) :
0 < G.modularRatio x y :=
div_pos (Nat.cast_pos.mpr (G.suborbitCard_pos hc hl x y))
(Nat.cast_pos.mpr (G.suborbitCard_pos hc hl y x))
lemma modularRatio_cocycle (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(x y z : V) : G.modularRatio x y * G.modularRatio y z = G.modularRatio x z := by
simp only [modularRatio_eq_relativeMass]
apply Subgroup.relativeMass_cocycle
all_goals
rw [←suborbitCard_eq_relIndex]
exact ne_of_gt (G.suborbitCard_pos hc hl _ _)
lemma modularRatio_self (x : V) : G.modularRatio x x = 1 := by
rw [modularRatio_eq_relativeMass,Subgroup.relativeMass_self]
lemma Aut.modularRatio (φ : G.Aut) (x y : V) :
G.modularRatio (φ.vertex x) (φ.vertex y) = G.modularRatio x y := by
simp only [CriticalPercolation.BondGraph.modularRatio,Aut.suborbitCard]
def CombinatoriallyUnimodular : Prop :=
∀ (φ : G.Aut) x, G.suborbitCard x (φ.vertex x) = G.suborbitCard (φ.vertex x) x
lemma CombinatoriallyUnimodular.modularRatio (hu : G.CombinatoriallyUnimodular)
(hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (φ : G.Aut) (x : V) :
G.modularRatio x (φ.vertex x) = 1 := by
simp only [CriticalPercolation.BondGraph.modularRatio,hu φ x,
div_self (ne_of_gt (Nat.cast_pos.mpr (G.suborbitCard_pos hc hl (φ.vertex x) x)) :
(G.suborbitCard (φ.vertex x) x : ℝ) ≠ 0)]
lemma CombinatoriallyUnimodular.rootWeight_invariant (hu : G.CombinatoriallyUnimodular)
(hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (o : V) (φ : G.Aut) (x : V) :
G.modularRatio o (φ.vertex x) = G.modularRatio o x := by
rw [← G.modularRatio_cocycle hc hl o (φ.vertex o) (φ.vertex x),
CombinatoriallyUnimodular.modularRatio G hu hc hl φ o,one_mul,Aut.modularRatio]
lemma modularRatio_card_balance (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(o x y : V) : G.modularRatio o x * (G.suborbitCard x y : ℝ) =
G.modularRatio o y * (G.suborbitCard y x : ℝ) := by
have hh := G.modularRatio_cocycle hc hl o x y
unfold modularRatio at hh
rw [← mul_div_assoc] at hh
exact (div_eq_iff (ne_of_gt (Nat.cast_pos.mpr (G.suborbitCard_pos hc hl y x)))).mp hh
end CriticalPercolation.BondGraph
end
end
end OAI
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