openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /LinearGroups /StrongDenseEigenvalues.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.LinearGroups.AbsoluteMatrixSpan | |
| import OAI.Probability.CriticalPercolation.LinearGroups.AllPlaceKronecker | |
| import OAI.Probability.CriticalPercolation.LinearGroups.ProperPadicModels | |
| import OAI.Probability.CriticalPercolation.LinearGroups.ProjectiveEigenvalues | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function Module | |
| open scoped Classical | |
| namespace CriticalPercolation.LinearGrowth | |
| variable {K L T G n : Type} [Field K] [Field L] [Field T] [Group G] | |
| variable [Fintype n] [DecidableEq n] | |
| variable {degree : ℕ} | |
| omit [DecidableEq n] in | |
| lemma eigenvector_transfer (f : K →+* T) (e : L →+* T) | |
| (A : Matrix n n K) (B : Matrix n n L) | |
| (hm : ∀ i j, f (A i j)=e (B i j)) | |
| (v : n → K) (w : n → L) (hv : ∀ i, f (v i)=e (w i)) | |
| (a : K) (b : L) (ha : f a=e b) (hAv : A.mulVec v=a • v) : B.mulVec w=b • w := by | |
| have hm' : A.map f=B.map e := by ext i j; exact hm i j | |
| have hv' : f ∘ v=e ∘ w := funext hv | |
| ext i | |
| apply e.injective | |
| rw [e.map_mulVec,←hm',←hv',←f.map_mulVec,hAv] | |
| simp only [Pi.smul_apply,smul_eq_mul,map_mul,ha,hv] | |
| omit [Fintype n] [DecidableEq n] in | |
| lemma eigenvector_transfer_ne_zero (f : K →+* T) (e : L →+* T) | |
| (v : n → K) (w : n → L) (hv : ∀ i, f (v i)=e (w i)) (hn : v≠0) : w≠0 := by | |
| intro hw | |
| apply hn | |
| ext i | |
| apply f.injective | |
| simpa only [hw,Pi.zero_apply,map_zero] using hv i | |
| lemma matrix_local_ratio_bound {K : Type} [NontriviallyNormedField K] [ProperSpace K] | |
| [DecidableEq G] (ρ : G →* GL n K) (hρ : StrongFullMatrixSpan ρ) | |
| (hg : SparseGrowth.HasPolynomialDegree G degree) (g : G) | |
| {v w : n → K} (hv : v≠0) (hw : w≠0) {a b : K} | |
| (ha : (ρ g).val.mulVec v=a • v) (hb : (ρ g).val.mulVec w=b • w) : ‖a/b‖≤1 := by | |
| let : Nontrivial (n → K) := ⟨⟨v,0,hv⟩⟩ | |
| obtain ⟨S,h1,hi,hgen,hgS⟩ := (SparseGrowth.hasPolynomialDegree_iff degree).mp hg | |
| obtain ⟨δ,hδ,hbnd⟩ := projective_bound_of_strongFullSpan (matrixRepresentation ρ) | |
| (hρ.strongFullSpan ρ) S h1 hgen degree hgS | |
| exact eigenvalue_ratio_norm_le_one (continuousRepresentation (matrixRepresentation ρ)) | |
| δ hδ hbnd g hv hw ha hb | |
| lemma eigenvalue_ratio_root_of_unity_of_strongFullMatrixSpan | |
| [CharZero K] [Countable K] [DecidableEq G] | |
| (ρ : G →* GL n K) (hρ : StrongFullMatrixSpan ρ) | |
| (hg : SparseGrowth.HasPolynomialDegree G degree) (g : G) | |
| {v w : n → K} (hv : v≠0) (hw : w≠0) {a b : K} | |
| (ha : (ρ g).val.mulVec v=a • v) (hb : (ρ g).val.mulVec w=b • w) : | |
| ∃ m : ℕ, 0 < m ∧ (a/b)^m=1 := by | |
| have ha0 : a≠0 := by | |
| intro he | |
| have hz : (matrixRepresentation ρ g) v=(matrixRepresentation ρ g) 0 := by | |
| simpa only [matrixRepresentation_apply,he,zero_smul,Matrix.mulVec_zero] using ha | |
| exact hv ((matrixRepresentation ρ g).injective hz) | |
| have hb0 : b≠0 := by | |
| intro he | |
| have hz : (matrixRepresentation ρ g) w=(matrixRepresentation ρ g) 0 := by | |
| simpa only [matrixRepresentation_apply,he,zero_smul,Matrix.mulVec_zero] using hb | |
| exact hw ((matrixRepresentation ρ g).injective hz) | |
| apply Arithmetic.root_of_unity_of_all_places (a/b) (div_ne_zero ha0 hb0) | |
| · intro f | |
| let τ := (Matrix.GeneralLinearGroup.map f).comp ρ | |
| have hm : ∀ g i j, f (ρ g i j)=(RingHom.id ℂ) (τ g i j) := fun _ _ _ => rfl | |
| have hτ := hρ.transfer f (RingHom.id ℂ) ρ τ hm | |
| let v' : n → ℂ := f ∘ v | |
| let w' : n → ℂ := f ∘ w | |
| have hv' : v'≠0 := eigenvector_transfer_ne_zero f (RingHom.id ℂ) v v' (fun _ => rfl) hv | |
| have hw' : w'≠0 := eigenvector_transfer_ne_zero f (RingHom.id ℂ) w w' (fun _ => rfl) hw | |
| have ha' := eigenvector_transfer f (RingHom.id ℂ) (ρ g).val (τ g).val | |
| (hm g) v v' (fun _ => rfl) a (f a) rfl ha | |
| have hb' := eigenvector_transfer f (RingHom.id ℂ) (ρ g).val (τ g).val | |
| (hm g) w w' (fun _ => rfl) b (f b) rfl hb | |
| simpa only [map_div₀] using matrix_local_ratio_bound τ hτ hg g hv' hw' ha' hb' | |
| · intro p hp f | |
| let : Fact p.Prime := hp | |
| let C := AlgebraicClosure (Padic p) | |
| let : Group.FG G := hg.fg | |
| let σ := (Matrix.GeneralLinearGroup.map f).comp ρ | |
| let E : Finset C := {f a,f b} ∪ Finset.univ.image (f ∘ v) ∪ Finset.univ.image (f ∘ w) | |
| obtain ⟨L,hL,hE,τ,hτ⟩ := Arithmetic.exists_finite_padic_matrix_model p σ E | |
| let : FiniteDimensional (Padic p) L := hL | |
| let : ProperSpace L := FiniteDimensional.proper (Padic p) L | |
| have haL : f a∈L := hE (by simp [E]) | |
| have hbL : f b∈L := hE (by simp [E]) | |
| have hvL (i : n) : f (v i)∈L := hE (by simp [E,Function.comp_def]) | |
| have hwL (i : n) : f (w i)∈L := hE (by simp [E,Function.comp_def]) | |
| let a' : L := ⟨f a,haL⟩ | |
| let b' : L := ⟨f b,hbL⟩ | |
| let v' : n → L := fun i => ⟨f (v i),hvL i⟩ | |
| let w' : n → L := fun i => ⟨f (w i),hwL i⟩ | |
| have hm : ∀ g i j, f (ρ g i j)=L.val.toRingHom (τ g i j) := by | |
| intro g i j | |
| exact (congrArg (fun M : GL n C => M i j) (hτ g)).symm | |
| have hτfull := hρ.transfer f L.val.toRingHom ρ τ hm | |
| have hv' : v'≠0 := eigenvector_transfer_ne_zero f L.val.toRingHom v v' (fun _ => rfl) hv | |
| have hw' : w'≠0 := eigenvector_transfer_ne_zero f L.val.toRingHom w w' (fun _ => rfl) hw | |
| have ha' := eigenvector_transfer f L.val.toRingHom (ρ g).val (τ g).val | |
| (hm g) v v' (fun _ => rfl) a a' rfl ha | |
| have hb' := eigenvector_transfer f L.val.toRingHom (ρ g).val (τ g).val | |
| (hm g) w w' (fun _ => rfl) b b' rfl hb | |
| have hh := matrix_local_ratio_bound τ hτfull hg g hv' hw' ha' hb' | |
| change ‖((a'/b' : L) : C)‖≤1 at hh | |
| change ‖f (a/b)‖≤1 | |
| simpa only [map_div₀, IntermediateField.coe_div] using hh | |
| end CriticalPercolation.LinearGrowth | |
| end | |
| end | |
| end OAI | |