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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 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 | 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
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