| import Mathlib |
| import OAI.Probability.CriticalPercolation.Growth.CosetSparseGrowth |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set Function |
| open scoped Classical Pointwise |
|
|
| namespace CriticalPercolation.RelativeGrowth |
| variable {A : Type*} [Group A] |
|
|
| lemma ball_smaller_stabilizer_bound (K L : Subgroup A) (hLK : L≤K) |
| [Finite (K ⧸ L.subgroupOf K)] (S : Set A) (n : ℕ) |
| (hf : (ball K S n).Finite) : |
| (ball L S n).Finite ∧ (ball L S n).ncard≤(ball K S n).ncard*Nat.card (K ⧸ L.subgroupOf K) := by |
| let e := Subgroup.quotientEquivProdOfLE hLK |
| have hsub : e '' ball L S n ⊆ ball K S n ×ˢ (Set.univ : Set (K ⧸ L.subgroupOf K)) := by |
| rintro _ ⟨_,⟨g,hg,rfl⟩,rfl⟩ |
| exact ⟨⟨g,hg,rfl⟩,Set.mem_univ _⟩ |
| have hprod := hf.prod (Set.finite_univ : (Set.univ : Set (K ⧸ L.subgroupOf K)).Finite) |
| have hfin : (ball L S n).Finite := (hprod.subset hsub).of_finite_image e.injective.injOn |
| refine ⟨hfin,?_⟩ |
| have hc := Set.ncard_le_ncard hsub hprod |
| rw [Set.ncard_image_of_injective _ e.injective,Set.ncard_prod,Set.ncard_univ] at hc |
| exact hc |
|
|
| lemma HasDegree.smaller_stabilizer {K L : Subgroup A} (hLK : L≤K) |
| [Finite (K ⧸ L.subgroupOf K)] {S : Set A} {d : ℕ} (hg : HasDegree K S d) : |
| HasDegree L S d := by |
| obtain ⟨hf,C,hC⟩ := hg |
| refine ⟨fun n => (ball_smaller_stabilizer_bound K L hLK S n (hf n)).1, |
| C*Nat.card (K ⧸ L.subgroupOf K),?_⟩ |
| intro R |
| obtain ⟨n,hn,hbound⟩ := hC R |
| refine ⟨n,hn,?_⟩ |
| have hh := (ball_smaller_stabilizer_bound K L hLK S n (hf n)).2 |
| calc |
| _ ≤ (ball K S n).ncard*Nat.card (K ⧸ L.subgroupOf K) := hh |
| _ ≤ (C*n^d)*Nat.card (K ⧸ L.subgroupOf K) := Nat.mul_le_mul_right _ hbound |
| _ = _ := by ring |
|
|
| end CriticalPercolation.RelativeGrowth |
| end |
| end |
|
|
| end OAI |
|
|
