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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 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphGeometry.GraphApproximateNeighborhoods
import OAI.Probability.CriticalPercolation.Growth.CosetBounded
namespace OAI
section
noncomputable section
open Set Function
open scoped Classical Pointwise
namespace CriticalPercolation.GraphTopology
variable {X : Type*} (Γ : SimpleGraph X)
def vertexStabilizer (o : X) : Subgroup (Γ ≃g Γ) where
carrier := {g | g o=o}
one_mem' := rfl
mul_mem' := by intro a b ha hb; change a (b o)=o; rw [hb,ha]
inv_mem' := by
intro a ha
change a.symm o=o
calc
a.symm o = a.symm (a o) := congrArg a.symm ha.symm
_ = o := a.symm_apply_apply o
@[simp] lemma mem_vertexStabilizer (o : X) (a : Γ ≃g Γ) : a∈vertexStabilizer Γ o ↔ a o=o := Iff.rfl
def stabilizerCosetEmbedding (o : X) : ((Γ ≃g Γ) ⧸ vertexStabilizer Γ o) ↪ X where
toFun := Quotient.lift (s:=QuotientGroup.leftRel (vertexStabilizer Γ o)) (fun g : Γ ≃g Γ => g o) (by
intro a b hab
have hab' : a⁻¹*b ∈ vertexStabilizer Γ o := (QuotientGroup.leftRel_apply).mp hab
change a.symm (b o)=o at hab'
have hh := congrArg a hab'
simpa using hh.symm)
inj' := Quotient.ind₂' (by
intro a b h
apply QuotientGroup.eq.mpr
change a.symm (b o)=o
change a o=b o at h
rw [←h]
exact a.symm_apply_apply o)
@[simp] lemma stabilizerCosetEmbedding_mk (o : X) (g : Γ ≃g Γ) :
stabilizerCosetEmbedding Γ o (QuotientGroup.mk g)=g o := rfl
lemma displacement_pow_subset (o : X) (n : ℕ) :
displacement Γ o 1^n⊆displacement Γ o n := by
induction n with
| zero =>
intro g hg
have he : g=1 := by simpa only [pow_zero,Set.mem_one] using hg
subst g
exact one_mem_displacement Γ o 0
| succ n ih =>
rw [pow_succ]
exact (Set.mul_subset_mul_right ih).trans (displacement_mul_subset Γ o n 1)
lemma stabilizer_coset_ball_image_subset (o : X) (n : ℕ) :
stabilizerCosetEmbedding Γ o '' RelativeGrowth.ball (vertexStabilizer Γ o) (displacement Γ o 1) n
⊆reachBall Γ o n := by
rintro _ ⟨_,⟨g,hg,rfl⟩,rfl⟩
exact displacement_pow_subset Γ o n hg
lemma displacement_relativeDegree (hl : ∀ x, (Γ.neighborSet x).Finite) (o : X) (d : ℕ)
(hpoly : ∀ R : ℕ, ∃ n≥R, (reachBall Γ o n).ncard≤n^d) :
RelativeGrowth.HasDegree (vertexStabilizer Γ o) (displacement Γ o 1) d := by
have hf (n : ℕ) : (RelativeGrowth.ball (vertexStabilizer Γ o) (displacement Γ o 1) n).Finite := by
apply Set.Finite.of_finite_image (f:=stabilizerCosetEmbedding Γ o)
· exact (reachBall_finite Γ hl o n).subset (stabilizer_coset_ball_image_subset Γ o n)
· exact (stabilizerCosetEmbedding Γ o).injective.injOn
refine ⟨hf,1,?_⟩
intro R
obtain ⟨n,hn,hb⟩ := hpoly R
refine ⟨n,hn,?_⟩
rw [one_mul]
apply le_trans _ hb
rw [←Set.ncard_image_of_injective _ (stabilizerCosetEmbedding Γ o).injective]
exact Set.ncard_le_ncard (stabilizer_coset_ball_image_subset Γ o n) (reachBall_finite Γ hl o n)
lemma vertexStabilizer_bounded (o : X) :
RelativeGrowth.Bounded (displacement Γ o 1) (vertexStabilizer Γ o:Set (Γ ≃g Γ)) := by
refine ⟨1,?_⟩
rw [pow_one]
intro g hg
change g o∈reachBall Γ o 1
rw [(mem_vertexStabilizer Γ o g).mp hg]
exact reachBall_self Γ o 1
end CriticalPercolation.GraphTopology
end
end
end OAI
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