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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 | 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
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