| 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 |
|
|
