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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 | import Mathlib
namespace OAI
section
noncomputable section
open Function
namespace CriticalPercolation.LinearGrowth
variable {K V G : Type*} [NontriviallyNormedField K] [NormedAddCommGroup V] [NormedSpace K V] [Group G]
lemma equiv_eigenvalue_ne_zero (T : V ≃L[K] V) {v : V} (hv : v≠0) {a : K}
(he : T v=a • v) : a≠0 := by
intro ha
apply hv
apply T.injective
simpa only [ha,zero_smul,map_zero] using he
lemma representation_pow_eigenvector (ρ : G →* (V ≃L[K] V)) (g : G)
{v : V} {a : K} (he : ρ g v=a • v) (n : ℕ) : ρ (g^n) v=a^n • v := by
induction n with
| zero => rw [pow_zero,map_one,pow_zero,one_smul]; rfl
| succ n ih =>
rw [pow_succ,map_mul]
change ρ (g^n) (ρ g v)=_
rw [he,map_smul,ih,smul_smul]
congr 1
rw [pow_succ,mul_comm]
lemma eigenvalue_ratio_norm_le_one (ρ : G →* (V ≃L[K] V))
(δ : ℝ) (hδ : 0<δ)
(hb : ∀ g : G, ∀ x : V, δ*‖(ρ g).toContinuousLinearMap‖*‖x‖≤‖ρ g x‖)
(g : G) {v w : V} (hv : v≠0) (hw : w≠0) {a b : K}
(ha : ρ g v=a • v) (hb' : ρ g w=b • w) : ‖a/b‖≤1 := by
have hb0 : b≠0 := equiv_eigenvalue_ne_zero _ hw hb'
have hvp : 0<‖v‖ := norm_pos_iff.mpr hv
have hwp : 0<‖w‖ := norm_pos_iff.mpr hw
have hbp : 0<‖b‖ := norm_pos_iff.mpr hb0
have hpow (n : ℕ) : ‖a/b‖^n≤1/δ := by
have hu := (ρ (g^n)).toContinuousLinearMap.le_opNorm v
change ‖ρ (g^n) v‖≤‖(ρ (g^n)).toContinuousLinearMap‖*‖v‖ at hu
rw [representation_pow_eigenvector ρ g ha n,norm_smul,norm_pow] at hu
have hu' : ‖a‖^n≤‖(ρ (g^n)).toContinuousLinearMap‖ := (mul_le_mul_iff_left₀ hvp).mp hu
have hl := hb (g^n) w
rw [representation_pow_eigenvector ρ g hb' n,norm_smul,norm_pow] at hl
have hl' : δ*‖(ρ (g^n)).toContinuousLinearMap‖≤‖b‖^n := (mul_le_mul_iff_left₀ hwp).mp hl
rw [norm_div,div_pow]
apply (div_le_div_iff₀ (pow_pos hbp n) hδ).mpr
nlinarith [mul_le_mul_of_nonneg_left hu' hδ.le]
by_contra hn
obtain ⟨n,hn'⟩ := pow_unbounded_of_one_lt (1/δ) (lt_of_not_ge hn)
exact hn'.not_ge (hpow n)
end CriticalPercolation.LinearGrowth
end
end
end OAI
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