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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Criticality.ExpandingCharacterGrowth
namespace OAI
section
noncomputable section
open Set Function
open scoped Classical Matrix
namespace CriticalPercolation.SparseGrowth
variable {A : Type*} [Group A] [DecidableEq A]
variable {ι : Type*} [Fintype ι] [DecidableEq ι]
def latticeCharacter (w : ι → ℂ) : Multiplicative (ι → ℤ) →* Multiplicative ℂ where
toFun x := Multiplicative.ofAdd (w ⬝ᵥ fun i => ((Multiplicative.toAdd x i : ℤ) : ℂ))
map_one' := by change Multiplicative.ofAdd _ = Multiplicative.ofAdd 0; congr 1; simp
map_mul' x y := by
change Multiplicative.ofAdd (w ⬝ᵥ fun i => (((Multiplicative.toAdd x i +
Multiplicative.toAdd y i) : ℤ) : ℂ)) = Multiplicative.ofAdd (_+_)
congr 1
simp [dotProduct,Finset.sum_add_distrib,mul_add]
omit [DecidableEq ι] in
lemma latticeCharacter_mulVec (w : ι → ℂ) (M : Matrix ι ι ℤ) (v : ι → ℤ) :
Multiplicative.toAdd (latticeCharacter w (Multiplicative.ofAdd (M.mulVec v))) =
(Matrix.vecMul w (M.map (Int.castRingHom ℂ))) ⬝ᵥ (fun i => (v i : ℂ)) := by
change (w ⬝ᵥ fun i => ((M.mulVec v i : ℤ) : ℂ)) = _
have he : (fun i => ((M.mulVec v i : ℤ) : ℂ)) =
(M.map (Int.castRingHom ℂ)).mulVec (fun i => (v i : ℂ)) := by
funext i
exact (Int.castRingHom ℂ).map_mulVec M v i
rw [he,Matrix.dotProduct_mulVec]
lemma latticeCharacter_single (w : ι → ℂ) (i : ι) :
Multiplicative.toAdd (latticeCharacter w (Multiplicative.ofAdd (Pi.single i 1)))=w i := by
change (w ⬝ᵥ fun j => (((Pi.single i (1:ℤ) : ι → ℤ) j : ℤ) : ℂ))=w i
have he : (fun j => (((Pi.single i (1:ℤ) : ι → ℤ) j : ℤ) : ℂ)) = Pi.single i (1:ℂ) := by
funext j
by_cases h : i=j
· subst j; simp
· simp [Ne.symm h]
rw [he,dotProduct_single,mul_one]
theorem HasPolynomialDegree.conjugationMatrix_roots_norm_le_one {d : ℕ}
(hg : HasPolynomialDegree A d) (H : Subgroup A) [H.Normal] (a : A)
(p : H →* Multiplicative (ι → ℤ)) (hp : Surjective p)
(M : Matrix ι ι ℤ)
(hM : ∀ x : H, Multiplicative.toAdd (p ((MulAut.conjNormal a) x)) =
M.mulVec (Multiplicative.toAdd (p x))) :
∀ z∈(M.map (Int.castRingHom ℂ)).charpoly.roots, ‖z‖≤1 := by
intro z hz
let C := M.map (Int.castRingHom ℂ)
have hroot : (C.transpose.mulVecLin.charpoly).IsRoot z := by
rw [Matrix.charpoly_mulVecLin,Matrix.charpoly_transpose]
exact (Polynomial.mem_roots (Matrix.charpoly_monic C).ne_zero).mp hz
have heig := (Module.End.hasEigenvalue_iff_isRoot_charpoly C.transpose.mulVecLin z).mpr hroot
obtain ⟨w,hw⟩ := heig.exists_hasUnifEigenvector
have hweq : Matrix.vecMul w C = z • w := by
simpa only [Matrix.mulVecLin_apply,Matrix.mulVec_transpose] using hw.apply_eq_smul
have hwn := hw.2
have hex : ∃ i, w i ≠ 0 := by
by_contra! h
exact hwn (funext h)
obtain ⟨i,hi⟩ := hex
obtain ⟨b,hb⟩ := hp (Multiplicative.ofAdd (Pi.single i 1))
let φ := (latticeCharacter w).comp p
have hφ : ∀ x : H, Multiplicative.toAdd (φ ((MulAut.conjNormal a) x)) =
z * Multiplicative.toAdd (φ x) := by
intro x
change Multiplicative.toAdd (latticeCharacter w (p ((MulAut.conjNormal a) x))) =
z * Multiplicative.toAdd (latticeCharacter w (p x))
have hpx : p ((MulAut.conjNormal a) x) =
Multiplicative.ofAdd (M.mulVec (Multiplicative.toAdd (p x))) := by
exact congrArg Multiplicative.ofAdd (hM x)
rw [hpx,latticeCharacter_mulVec]
change (Matrix.vecMul w C) ⬝ᵥ (fun i => ((Multiplicative.toAdd (p x) i : ℤ) : ℂ)) = _
rw [hweq,smul_dotProduct]
rfl
have hφb : Multiplicative.toAdd (φ b) ≠ 0 := by
change Multiplicative.toAdd (latticeCharacter w (p b)) ≠ 0
rw [hb,latticeCharacter_single]
exact hi
exact hg.norm_conj_eigencharacter_le_one H a φ z hφ b hφb
end CriticalPercolation.SparseGrowth
end
end
end OAI
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