openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Gaussian /GaussianGramUniformIntegrability.lean
| 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 | |