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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 42 43 44 45 46 47 48 49 50 51 52 53 54 55 | import OAI.Probability.InvariantIsing.Gaussian.GaussianGramNormBound
import OAI.Probability.InvariantIsing.Gaussian.GaussianSingularSquareUI
namespace OAI
/-! The Davidson--Szarek uniform-integrability consequence for the physical Gram spectrum. -/
noncomputable section
open MeasureTheory ProbabilityTheory
namespace InvariantIsing
lemma uniformIntegrable_nat_mul_real {Ω : Type*} [MeasurableSpace Ω]
(P : Measure Ω) [IsProbabilityMeasure P] (X : ℕ → Ω → ℝ)
(hX : UniformIntegrable X 1 P) (n : ℕ) :
UniformIntegrable (fun k ω => (n : ℝ)*X k ω) 1 P := by
induction n with
| zero => simpa only [Nat.cast_zero,zero_mul] using
(uniformIntegrable_const (ι := ℕ) le_rfl (by simp) (memLp_const (0 : ℝ)))
| succ n ih =>
simpa only [Nat.cast_add,Nat.cast_one,add_mul,one_mul] using
uniformIntegrable_add_real P _ _ ih hX
theorem gaussianPattern_spectralRadius_uniformIntegrable {α : ℝ} (hα : 0 ≤ α)
{Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
(Z : (N : ℕ) → Ω → EuclideanSpace ℝ (Fin N × Fin (gaussianPatternCount α N)))
(hZ : ∀ N, HasLaw (Z N) (stdGaussian _) P) (c : ℝ) :
UniformIntegrable (fun k ω => spectralRadius
(fun i => c*gaussianPatternEigenvalues (Z (k+1) ω) i)) 1 P := by
obtain ⟨n,hn⟩ := exists_nat_gt |c|
let X (k : ℕ) (ω : Ω) := (gaussianPatternSingularMax (Z (k+1) ω))^2/(k+1)
have hX : UniformIntegrable X 1 P := gaussianPatternSingularMax_square_uniformIntegrable hα P Z hZ
refine uniformIntegrable_of_abs_le P _ _ ?_ (uniformIntegrable_nat_mul_real P X hX n) ?_
· intro k
have he : Measurable (fun z : EuclideanSpace ℝ (Fin (k+1) × Fin (gaussianPatternCount α (k+1))) =>
spectralRadius (fun i => c*gaussianPatternEigenvalues z i)) := by
apply measurable_spectralRadius
exact Measurable.of_eval (fun i =>
((measurable_pi_apply i).comp
(measurable_gaussianPatternEigenvalues (k+1) (gaussianPatternCount α (k+1)))).const_mul c)
exact (he.aemeasurable.comp_aemeasurable (hZ (k+1)).aemeasurable).aestronglyMeasurable
intro k
apply ae_of_all
intro ω
have hpos : 0 ≤ X k ω := div_nonneg (sq_nonneg _) (by positivity)
have hr : 0 ≤ spectralRadius (fun i => c*gaussianPatternEigenvalues (Z (k+1) ω) i) :=
(abs_nonneg _).trans (abs_le_spectralRadius _ (0 : Fin (k+1)))
rw [abs_of_nonneg hr,abs_of_nonneg (mul_nonneg (Nat.cast_nonneg n) hpos)]
exact (gaussianPattern_spectralRadius_le c (Z (k+1) ω)).trans
(by rw [mul_div_assoc]; exact mul_le_mul_of_nonneg_right hn.le hpos)
end InvariantIsing
end
end OAI
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