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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 | 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
end OAI
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