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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 56 57 58 59 60 61 62 63 64 | import OAI.Probability.InvariantIsing.Spectral.PositiveResolventSpectrum
namespace OAI
/-! Uniform trace and quadratic bounds for positive resolvents. -/
noncomputable section
open Matrix
open scoped BigOperators RealInnerProductSpace
namespace InvariantIsing
lemma positiveResolvent_trace_le {N : ℕ} {t : ℝ} (ht : 0 < t)
{B : Matrix (Fin N) (Fin N) ℝ} (hB : B.PosSemidef) :
(positiveResolvent t B).trace ≤ (N : ℝ)/t := by
have he := (positiveResolvent_posDef ht hB).isHermitian.trace_eq_sum_eigenvalues
simp only [RCLike.ofReal_real_eq_id,id_eq] at he
rw [he]
calc
_ ≤ ∑ _ : Fin N, 1/t := Finset.sum_le_sum (fun i _ => (positiveResolvent_eigenvalues ht hB i).2)
_ = _ := by simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]; ring
lemma positiveResolvent_quadratic_le {N : ℕ} {t : ℝ} (ht : 0 < t)
{B : Matrix (Fin N) (Fin N) ℝ} (hB : B.PosSemidef) (x : EuclideanSpace ℝ (Fin N)) :
gaussianQuadraticForm (positiveResolvent t B) x ≤ ‖x‖^2/t := by
let hR := (positiveResolvent_posDef ht hB).isHermitian
rw [gaussianQuadraticForm,hermitian_quadratic_coordinates _ hR]
calc
_ ≤ ∑ i, (1/t)*(hR.eigenvectorBasis.repr x i)^2 :=
Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_right
(positiveResolvent_eigenvalues ht hB i).2 (sq_nonneg _))
_ = (1/t)*‖hR.eigenvectorBasis.repr x‖^2 := by
rw [EuclideanSpace.real_norm_sq_eq,Finset.mul_sum]
_ = _ := by rw [LinearIsometryEquiv.norm_map]; ring
lemma positiveResolvent_quadratic_nonneg {N : ℕ} {t : ℝ} (ht : 0 < t)
{B : Matrix (Fin N) (Fin N) ℝ} (hB : B.PosSemidef) (x : EuclideanSpace ℝ (Fin N)) :
0 ≤ gaussianQuadraticForm (positiveResolvent t B) x := by
rw [gaussianQuadraticForm,EuclideanSpace.inner_eq_star_dotProduct]
simp only [star_trivial]
rw [dotProduct_comm]
exact (positiveResolvent_posDef ht hB).posSemidef.dotProduct_mulVec_nonneg x.ofLp
lemma positiveResolvent_normalized_abs_bound {N : ℕ} (hN : 0 < N) {t : ℝ} (ht : 0 < t)
{B : Matrix (Fin N) (Fin N) ℝ} (hB : B.PosSemidef) (x : EuclideanSpace ℝ (Fin N)) :
|(1/(N : ℝ))*gaussianQuadraticForm (positiveResolvent t B) x-
(positiveResolvent t B).trace/N| ≤ (1/((N : ℝ)*t))*‖x‖^2+1/t := by
have hn : (0 : ℝ) < N := by exact_mod_cast hN
have hq := positiveResolvent_quadratic_nonneg ht hB x
have htr := (positiveResolvent_posDef ht hB).posSemidef.trace_nonneg
calc
_ ≤ |(1/(N : ℝ))*gaussianQuadraticForm (positiveResolvent t B) x|+
|(positiveResolvent t B).trace/N| := abs_sub _ _
_ = (1/(N : ℝ))*gaussianQuadraticForm (positiveResolvent t B) x+
(positiveResolvent t B).trace/N := by rw [abs_of_nonneg (by positivity),abs_of_nonneg (by positivity)]
_ ≤ (1/(N : ℝ))*(‖x‖^2/t)+((N : ℝ)/t)/N := add_le_add
(mul_le_mul_of_nonneg_left (positiveResolvent_quadratic_le ht hB x) (by positivity))
(div_le_div_of_nonneg_right (positiveResolvent_trace_le ht hB) hn.le)
_ = _ := by field_simp
end InvariantIsing
end
end OAI
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