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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.Spectral.PositiveResolventSpectrum
namespace OAI
/-! The dimension-explicit Gaussian quadratic variance bound for a positive resolvent. -/
noncomputable section
open MeasureTheory ProbabilityTheory Matrix
open scoped BigOperators
namespace InvariantIsing
lemma positiveResolvent_quadratic_variance {N : ℕ} {t : ℝ} (ht : 0 < t)
{B : Matrix (Fin N) (Fin N) ℝ} (hB : B.PosSemidef) :
variance (gaussianQuadraticForm (positiveResolvent t B)) (stdGaussian _) ≤ 2*N/t^2 := by
rw [hermitianGaussianQuadratic_variance _ (positiveResolvent_posDef ht hB).isHermitian]
have hs : (∑ i, ((positiveResolvent_posDef ht hB).isHermitian.eigenvalues i)^2) ≤
(N : ℝ)*(1/t)^2 := by
calc
_ ≤ ∑ _ : Fin N, (1/t)^2 := Finset.sum_le_sum (fun i _ =>
(sq_le_sq₀ (positiveResolvent_eigenvalues ht hB i).1 (by positivity)).2
(positiveResolvent_eigenvalues ht hB i).2)
_ = _ := by simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]
calc
_ ≤ 2*((N : ℝ)*(1/t)^2) := mul_le_mul_of_nonneg_left hs (by norm_num)
_ = _ := by ring
lemma positiveResolvent_normalized_variance {N : ℕ} (hN : 0 < N) {t : ℝ} (ht : 0 < t)
{B : Matrix (Fin N) (Fin N) ℝ} (hB : B.PosSemidef) :
variance (fun x => (1/(N : ℝ))*gaussianQuadraticForm (positiveResolvent t B) x) (stdGaussian _) ≤
2/((N : ℝ)*t^2) := by
rw [variance_const_mul]
have h := mul_le_mul_of_nonneg_left (positiveResolvent_quadratic_variance ht hB) (sq_nonneg (1/(N : ℝ)))
apply h.trans_eq
have hn : (N : ℝ) ≠ 0 := by exact_mod_cast Nat.ne_of_gt hN
field_simp [hn]
end InvariantIsing
end
end OAI
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