File size: 3,455 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 | import OAI.Probability.InvariantIsing.Haar.HaarExponentialGradient
namespace OAI
/-! Dominated transport from positive polynomial truncations to exponential observables. -/
noncomputable section
open Matrix MvPolynomial MeasureTheory Filter Set
open scoped Topology
namespace InvariantIsing
lemma haarExpPolynomial_test_integral_tendsto {N : ℕ} (μ : Measure (SpecialOrthogonal N))
[IsFiniteMeasure μ] (p : MatrixPolynomial N)
(hp : ∀ U : SpecialOrthogonal N, 0 ≤ haarPolynomialValue p U)
(F : ℝ → ℝ) (hF : Continuous F) :
Tendsto (fun n : ℕ => ∫ U, F (haarPolynomialValue (haarExpPolynomial (n+1) p) U) ∂μ)
atTop (𝓝 (∫ U, F (Real.exp (haarPolynomialValue p U)) ∂μ)) := by
obtain ⟨B,hB⟩ := haarPolynomialValue_bound p
obtain ⟨K,hK⟩ := isCompact_Icc.exists_bound_of_continuousOn
(show ContinuousOn F (Icc 1 (Real.exp B)) from hF.continuousOn)
apply tendsto_integral_of_dominated_convergence (fun _ => K)
(fun n => (hF.comp (continuous_haarPolynomialValue _)).aestronglyMeasurable)
(integrable_const K)
· intro n
apply ae_of_all
intro U
apply hK
refine ⟨haarExpPolynomial_positive (n+1) p _ (hp U),?_⟩
rw [haarPolynomialValue,haarExpPolynomial_eval]
exact (positiveExpSum_le_exp (n+1) (hp U)).trans
(Real.exp_le_exp.mpr ((le_abs_self _).trans (hB U)))
· apply ae_of_all
intro U
apply hF.continuousAt.tendsto.comp
have h := (positiveExpSum_tendsto (haarPolynomialValue p U)).comp (tendsto_add_atTop_nat 1)
simpa only [Function.comp_def,haarPolynomialValue,haarExpPolynomial_eval,positiveExpSum] using h
theorem haar_exp_polynomial_entropy_bound {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) :
(∫ U, Real.exp (haarPolynomialValue p U)*Real.log (Real.exp (haarPolynomialValue p U)) ∂μ)-
(∫ U, Real.exp (haarPolynomialValue p U) ∂μ)*
Real.log (∫ U, Real.exp (haarPolynomialValue p U) ∂μ) ≤
C*(∫ U, Real.exp (haarPolynomialValue p U) ∂μ)/(2*((N:ℝ)-2)) := by
have hn (n : ℕ) :
(∫ U, haarPolynomialValue (haarExpPolynomial (n+1) p) U*
Real.log (haarPolynomialValue (haarExpPolynomial (n+1) p) U) ∂μ)-
(∫ U, haarPolynomialValue (haarExpPolynomial (n+1) p) U ∂μ)*
Real.log (∫ U, haarPolynomialValue (haarExpPolynomial (n+1) p) U ∂μ) ≤
C*(∫ U, haarPolynomialValue (haarExpPolynomial (n+1) p) U ∂μ)/(2*((N:ℝ)-2)) := by
obtain ⟨d,hd⟩ := exists_mem_haarPolynomialSpace (haarExpPolynomial (n+1) p)
exact haarPolynomial_entropy_bound hN μ ⟨_,hd⟩ 1 C zero_lt_one
(fun U => haarExpPolynomial_positive (n+1) p _ (hp U))
(fun U => haarExpPolynomial_log_gradient_bound n p _ (hp U) C hC (hG U))
have hm := haarExpPolynomial_test_integral_tendsto μ p hp id continuous_id
simp only [id_eq] at hm
have he := haarExpPolynomial_test_integral_tendsto μ p hp (fun x => x*Real.log x) Real.continuous_mul_log
exact le_of_tendsto_of_tendsto (he.sub (Real.continuous_mul_log.continuousAt.tendsto.comp hm))
((hm.const_mul C).div_const (2*((N:ℝ)-2))) (Eventually.of_forall hn)
end InvariantIsing
end
end OAI
|