| import OAI.Probability.InvariantIsing.Pressure.RandomSpectralBound |
| import OAI.Probability.InvariantIsing.Spectral.SpectralExcess |
|
|
| namespace OAI |
|
|
| /-! Turning a scalar edge-excess limit into the selected spectral hypotheses. -/ |
| noncomputable section |
| open Filter Set |
| open scoped Topology |
| namespace InvariantIsing |
|
|
| lemma selected_no_outliers_of_excess (N : ℕ → ℕ) (eig : (k : ℕ) → Fin (N k) → ℝ) |
| (a b : ℝ) (he : Tendsto (fun k => spectralExcess (eig k) a b) atTop (𝓝 0)) : |
| ∀ ε : ℝ, 0 < ε → ∀ᶠ k in atTop, ∀ i, a-ε ≤ eig k i ∧ eig k i ≤ b+ε := by |
| intro ε hε |
| filter_upwards [he.eventually (Iio_mem_nhds hε)] with k hk i |
| have hb := spectralExcess_bounds (eig k) a b i |
| constructor <;> linarith |
|
|
| lemma selected_spectralRadius_bounded (s : ℕ → ℕ) |
| (eig : (k : ℕ) → Fin (s k+1) → ℝ) (a b : ℝ) |
| (he : Tendsto (fun k => spectralExcess (eig k) a b) atTop (𝓝 0)) : |
| ∃ K : ℝ, ∀ᶠ k in atTop, spectralRadius (eig k) ≤ K := by |
| refine ⟨max |a| |b|+1,?_⟩ |
| filter_upwards [selected_no_outliers_of_excess (fun k => s k+1) eig a b he 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 |
|
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| end OAI |
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|
