File size: 3,146 Bytes
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 65 66 67 68 69 | import OAI.Probability.InvariantIsing.Haar.HaarPolynomialMoment
namespace OAI
/-! Gaussian exponential-moment bounds for nonnegative polynomial observables. -/
noncomputable section
open Matrix MvPolynomial MeasureTheory
open scoped BigOperators
namespace InvariantIsing
lemma haarPolynomialGamma_scaled_value {N : ℕ} (p : MatrixPolynomial N) (c : ℝ)
(M : Matrix (Fin N) (Fin N) ℝ) :
matrixPolynomialEval M (haarPolynomialGamma (c • p) (c • p)) =
c^2*matrixPolynomialEval M (haarPolynomialGamma p p) := by
simp only [haarPolynomialGamma,Derivation.map_smul_of_tower,map_sum,map_mul,map_smul,
smul_eq_mul,Finset.mul_sum]
apply Finset.sum_congr rfl
intro i _
apply Finset.sum_congr rfl
intro j _
ring
theorem haarPolynomialMoment_log_bound_of_nonneg {N : ℕ} (hN : 3 ≤ N)
(μ : Measure (SpecialOrthogonal N)) [IsProbabilityMeasure μ] [μ.IsMulLeftInvariant]
(p : MatrixPolynomial N) (hp : ∀ U : SpecialOrthogonal N, 0 ≤ haarPolynomialValue p U)
(C : ℝ) (hC : 0 ≤ C)
(hG : ∀ U : SpecialOrthogonal N, haarPolynomialValue (haarPolynomialGamma p p) U ≤ C)
{t : ℝ} (ht : 0 < t) :
Real.log (haarPolynomialMoment μ p t) ≤
(∫ U, haarPolynomialValue p U ∂μ)*t+C/(2*((N:ℝ)-2))*t^2 := by
have hE (s : ℝ) (hs : 0 < s) :
s*haarPolynomialMomentDerivative μ p s-
haarPolynomialMoment μ p s*Real.log (haarPolynomialMoment μ p s) ≤
(C/(2*((N:ℝ)-2)))*s^2*haarPolynomialMoment μ p s := by
have hscaled (U : SpecialOrthogonal N) : haarPolynomialValue (s • p) U = s*haarPolynomialValue p U := by
simp only [haarPolynomialValue,map_smul,smul_eq_mul]
have hp' (U : SpecialOrthogonal N) : 0 ≤ haarPolynomialValue (s • p) U := by
rw [hscaled]
exact mul_nonneg hs.le (hp U)
have hG' (U : SpecialOrthogonal N) :
haarPolynomialValue (haarPolynomialGamma (s • p) (s • p)) U ≤ C*s^2 := by
rw [haarPolynomialValue,haarPolynomialGamma_scaled_value]
have hh := mul_le_mul_of_nonneg_left (hG U) (sq_nonneg s)
simpa only [haarPolynomialValue,mul_comm] using hh
have he := haar_exp_polynomial_entropy_bound hN μ (s • p) hp' (C*s^2)
(mul_nonneg hC (sq_nonneg s)) hG'
simp_rw [hscaled,Real.log_exp] at he
have hi : (∫ U, Real.exp (s*haarPolynomialValue p U)*(s*haarPolynomialValue p U) ∂μ) =
s*haarPolynomialMomentDerivative μ p s := by
rw [haarPolynomialMomentDerivative,← integral_const_mul]
apply integral_congr_ae
exact ae_of_all μ fun U => by ring
rw [hi] at he
change s*haarPolynomialMomentDerivative μ p s-
haarPolynomialMoment μ p s*Real.log (haarPolynomialMoment μ p s) ≤
C*s^2*haarPolynomialMoment μ p s/(2*((N:ℝ)-2)) at he
linear_combination he
have hh := log_moment_le_of_entropy (haarPolynomialMoment μ p)
(haarPolynomialMomentDerivative μ p) (C/(2*((N:ℝ)-2)))
(haarPolynomialMoment_pos μ p) (haarPolynomialMoment_zero μ p)
(haarPolynomialMoment_hasDerivAt μ p) hE ht
simpa only [haarPolynomialMomentDerivative_zero] using hh
end InvariantIsing
end
end OAI
|