| import Mathlib |
| import OAI.Probability.CriticalPercolation.GraphTopology.GraphGeneratingNeighborhood |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set Function Topology |
| open scoped Classical Pointwise |
|
|
| namespace CriticalPercolation.GraphTopology |
| variable {X : Type*} (Γ : SimpleGraph X) |
|
|
| lemma image_displacement_ncard_le {B : Type*} [Group B] |
| (hl : ∀ x, (Γ.neighborSet x).Finite) |
| (ht : ∀ u v, ∃ g : Γ ≃g Γ, g u=v) (o : X) |
| (f : (Γ ≃g Γ) →* B) |
| (hB : (f '' {g : Γ ≃g Γ | g o=o}).Finite) (r : ℕ) : |
| (f '' displacement Γ o r).ncard ≤ |
| (reachBall Γ o r).ncard * (f '' {g : Γ ≃g Γ | g o=o}).ncard := by |
| let a : X → (Γ ≃g Γ) := fun x => (ht o x).choose |
| have ha : ∀ x, a x o=x := fun x => (ht o x).choose_spec |
| let U := reachBall Γ o r ×ˢ (f '' {g : Γ ≃g Γ | g o=o}) |
| let m : X × B → B := fun z => f (a z.1)*z.2 |
| have hU : U.Finite := (reachBall_finite Γ hl o r).prod hB |
| have hs : f '' displacement Γ o r ⊆ m '' U := by |
| rintro b ⟨g,hg,rfl⟩ |
| refine ⟨(g o,f ((a (g o))⁻¹*g)),⟨hg,?_⟩,?_⟩ |
| · refine ⟨(a (g o))⁻¹*g,?_,rfl⟩ |
| change (a (g o)).symm (g o)=o |
| calc |
| (a (g o)).symm (g o) = (a (g o)).symm ((a (g o)) o) := |
| congrArg (a (g o)).symm (ha (g o)).symm |
| _ = o := (a (g o)).symm_apply_apply o |
| · dsimp only [m] |
| rw [←map_mul,mul_inv_cancel_left] |
| calc |
| (f '' displacement Γ o r).ncard ≤ (m '' U).ncard := |
| Set.ncard_le_ncard hs (hU.image m) |
| _ ≤ U.ncard := Set.ncard_image_le hU |
| _ = (reachBall Γ o r).ncard * (f '' {g : Γ ≃g Γ | g o=o}).ncard := |
| Set.ncard_prod |
|
|
| theorem quotient_powers_ncard_le |
| [TopologicalSpace X] [DiscreteTopology X] |
| (hc : Γ.Connected) (hl : ∀ x, (Γ.neighborSet x).Finite) |
| (ht : ∀ u v, ∃ g : Γ ≃g Γ, g u=v) (o : X) |
| (H : Subgroup (Γ ≃g Γ)) [H.Normal] |
| (ho : @IsOpen (Γ ≃g Γ) (autTopology Γ) (H : Set (Γ ≃g Γ))) : |
| let T := (QuotientGroup.mk' H) '' displacement Γ o 1 |
| ∃ M : ℕ, 1≤M ∧ T.Finite ∧ (1 : (Γ ≃g Γ) ⧸ H)∈T ∧ T⁻¹=T ∧ |
| Subgroup.closure T=⊤ ∧ ∀ n : ℕ, 1≤n → |
| (T^n).ncard ≤ (reachBall Γ o n).ncard*M := by |
| let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ |
| let _ : DiscreteTopology ((Γ ≃g Γ) ⧸ H) := QuotientGroup.discreteTopology ho |
| let f := QuotientGroup.mk' H |
| let B := f '' {g : Γ ≃g Γ | g o=o} |
| have hB : B.Finite := isCompact_iff_finite.mp |
| ((compact_stabilizer Γ hc hl o).image QuotientGroup.continuous_mk) |
| have hB1 : (1 : (Γ ≃g Γ) ⧸ H)∈B := ⟨1,rfl,map_one f⟩ |
| have hM : 1≤B.ncard := (Set.ncard_pos hB).mpr ⟨1,hB1⟩ |
| dsimp only |
| refine ⟨B.ncard,hM,?_,⟨1,one_mem_displacement Γ o 1,map_one f⟩,?_,?_,?_⟩ |
| · exact isCompact_iff_finite.mp |
| ((compact_displacement Γ hc hl o 1).image QuotientGroup.continuous_mk) |
| · rw [←Set.image_inv] |
| congr 1 |
| ext g |
| exact displacement_inv Γ |
| · rw [←MonoidHom.map_closure,displacement_generates Γ hc ht o] |
| exact Subgroup.map_top_of_surjective _ QuotientGroup.mk_surjective |
| · intro n hn |
| rw [←Set.image_pow,←show displacement Γ o n=displacement Γ o 1^n by |
| simpa only [Nat.sub_add_cancel hn] using displacement_eq_pow Γ ht o (n-1)] |
| exact image_displacement_ncard_le Γ hl ht o f hB n |
|
|
| end CriticalPercolation.GraphTopology |
| end |
| end |
|
|
| end OAI |
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|
