openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Spectral /PositiveResolventFluctuation.lean
| import OAI.Probability.InvariantIsing.Spectral.PositiveResolventVariance | |
| import OAI.Probability.InvariantIsing.Magnetic.RestrictedPressureFluctuation | |
| namespace OAI | |
| /-! Uniform Gaussian mean absolute fluctuations of a normalized resolvent quadratic form. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Matrix | |
| namespace InvariantIsing | |
| lemma positiveResolvent_normalized_mean {N : ℕ} {t : ℝ} (ht : 0 < t) | |
| {B : Matrix (Fin N) (Fin N) ℝ} (hB : B.PosSemidef) : | |
| (∫ x, (1/(N : ℝ))*gaussianQuadraticForm (positiveResolvent t B) x ∂stdGaussian _) = | |
| (positiveResolvent t B).trace / N := by | |
| rw [integral_const_mul,hermitianGaussianQuadratic_mean _ (positiveResolvent_posDef ht hB).isHermitian] | |
| ring | |
| lemma positiveResolvent_normalized_fluctuation {N : ℕ} (hN : 0 < N) {t : ℝ} (ht : 0 < t) | |
| {B : Matrix (Fin N) (Fin N) ℝ} (hB : B.PosSemidef) : | |
| (∫ x, |(1/(N : ℝ))*gaussianQuadraticForm (positiveResolvent t B) x- | |
| (positiveResolvent t B).trace / N| ∂stdGaussian _) ≤ Real.sqrt (2/((N : ℝ)*t^2)) := by | |
| have hf := (gaussianQuadraticForm_memLp (positiveResolvent t B)).const_mul (1/(N : ℝ)) | |
| have h := integral_abs_sub_mean_le_sqrt_variance (stdGaussian _) | |
| (fun x => (1/(N : ℝ))*gaussianQuadraticForm (positiveResolvent t B) x) hf | |
| rw [positiveResolvent_normalized_mean ht hB] at h | |
| exact h.trans (Real.sqrt_le_sqrt (positiveResolvent_normalized_variance hN ht hB)) | |
| end InvariantIsing | |
| end | |
| end OAI | |