| import OAI.Probability.InvariantIsing.Haar.HaarPolynomialMoment |
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| namespace OAI |
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| /-! Gaussian exponential-moment bounds for nonnegative polynomial observables. -/ |
| noncomputable section |
| open Matrix MvPolynomial MeasureTheory |
| open scoped BigOperators |
| namespace InvariantIsing |
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| 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 |
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| 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 |
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| end InvariantIsing |
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| end |
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| end OAI |
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