| import OAI.Probability.InvariantIsing.Spectral.SpectralExtremeLimits |
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| namespace OAI |
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| /-! The extreme eigenvalues control a random spectrum without moment hypotheses. -/ |
| noncomputable section |
| open Filter Set |
| open scoped Topology |
| namespace InvariantIsing |
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| def spectralRadius {n : ℕ} (eig : Fin (n+1) → ℝ) : ℝ := |
| max |spectralMinimum eig| |spectralMaximum eig| |
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| 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 _ _)) |
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| 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 |
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| 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 |
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| 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] |
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| end InvariantIsing |
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| end |
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| end OAI |
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