File size: 2,522 Bytes
a89d282 | 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 | import OAI.Probability.InvariantIsing.Spectral.SpectralExtremeLimits
namespace OAI
/-! The extreme eigenvalues control a random spectrum without moment hypotheses. -/
noncomputable section
open Filter Set
open scoped Topology
namespace InvariantIsing
def spectralRadius {n : ℕ} (eig : Fin (n+1) → ℝ) : ℝ :=
max |spectralMinimum eig| |spectralMaximum eig|
lemma abs_le_spectralRadius {n : ℕ} (eig : Fin (n+1) → ℝ) (i : Fin (n+1)) :
|eig i| ≤ spectralRadius eig := by
apply abs_le.mpr
constructor
· exact (neg_le_neg (le_max_left _ _)).trans
((neg_abs_le (spectralMinimum eig)).trans (spectralMinimum_le eig i))
· exact (le_spectralMaximum eig i).trans
((le_abs_self _).trans (le_max_right _ _))
lemma spectralRadius_le_iff {n : ℕ} (eig : Fin (n+1) → ℝ) (K : ℝ) :
spectralRadius eig ≤ K ↔ ∀ i, |eig i| ≤ K := by
constructor
· intro h i
exact (abs_le_spectralRadius eig i).trans h
· intro h
apply max_le
· obtain ⟨i,_,hi⟩ := Finset.mem_image.mp (Finset.min'_mem
(Finset.univ.image eig) (Finset.image_nonempty.mpr Finset.univ_nonempty))
simpa only [spectralMinimum,← hi] using h i
· obtain ⟨i,_,hi⟩ := Finset.mem_image.mp (Finset.max'_mem
(Finset.univ.image eig) (Finset.image_nonempty.mpr Finset.univ_nonempty))
simpa only [spectralMaximum,← hi] using h i
lemma spectralRadius_eventually_bounded (eig : (N : ℕ) → Fin N → ℝ) (a b : ℝ)
(hmin : Tendsto (fun k => spectralMinimum (eig (k+1))) atTop (𝓝 a))
(hmax : Tendsto (fun k => spectralMaximum (eig (k+1))) atTop (𝓝 b)) :
∃ K : ℝ, ∀ᶠ k in atTop, spectralRadius (eig (k+1)) ≤ K := by
refine ⟨max |a| |b|+1,?_⟩
have ht := hmin.abs.max hmax.abs
exact (ht.eventually (Iio_mem_nhds (lt_add_one _))).mono fun _ h => h.le
lemma spectralRadius_eventually_bounded_of_no_outliers
(eig : (N : ℕ) → Fin N → ℝ) (a b : ℝ)
(hno : ∀ ε : ℝ, 0 < ε → ∀ᶠ N in atTop, ∀ i, a-ε ≤ eig N i ∧ eig N i ≤ b+ε) :
∃ K : ℝ, ∀ᶠ k in atTop, spectralRadius (eig (k+1)) ≤ K := by
refine ⟨max |a| |b|+1,?_⟩
filter_upwards [(tendsto_add_atTop_nat 1).eventually (hno 1 zero_lt_one)] with k hk
apply (spectralRadius_le_iff _ _).mpr
intro i
apply abs_le.mpr
have ha := neg_abs_le a
have hb := le_abs_self b
have hma := le_max_left |a| |b|
have hmb := le_max_right |a| |b|
constructor <;> linarith [(hk i).1,(hk i).2]
end InvariantIsing
end
end OAI
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