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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 | import OAI.Probability.InvariantIsing.Haar.HaarHeatOscillation
import OAI.Probability.InvariantIsing.Haar.HaarHeatMean
import OAI.Probability.InvariantIsing.Haar.HaarHeatOrder
namespace OAI
/-! Convergence of the actual orthogonal polynomial heat flow to its Haar mean. -/
noncomputable section
open Matrix MvPolynomial MeasureTheory Filter
open scoped Topology
namespace InvariantIsing
theorem haarPolynomialHeat_tendsto_mean {N d : ℕ} (hN : 3 ≤ N)
(μ : Measure (SpecialOrthogonal N)) [IsProbabilityMeasure μ] [μ.IsMulLeftInvariant]
(p : haarPolynomialSpace N d) (U : SpecialOrthogonal N) :
Tendsto (fun t : ℝ => haarPolynomialValue
((haarPolynomialHeat N d t p : haarPolynomialSpace N d) : MatrixPolynomial N) U)
atTop (𝓝 (∫ V, haarPolynomialValue (p : MatrixPolynomial N) V ∂μ)) := by
let u (t : ℝ) (V : SpecialOrthogonal N) := haarPolynomialValue
((haarPolynomialHeat N d t p : haarPolynomialSpace N d) : MatrixPolynomial N) V
obtain ⟨B,hB⟩ := haarPolynomialValue_bound (p : MatrixPolynomial N)
have hub (t : ℝ) (ht : 0 ≤ t) (V : SpecialOrthogonal N) : |u t V| ≤ B := by
apply abs_le.mpr
apply haarPolynomialHeat_bounds p (-B) B
· intro W
exact (abs_le.mp (hB W)).1
· intro W
exact (abs_le.mp (hB W)).2
· exact ht
have hlim : Tendsto (fun t : ℝ => ∫ V, u t V-u t U ∂μ) atTop (𝓝 0) := by
have h := tendsto_integral_filter_of_dominated_convergence
(μ := μ) (l := atTop) (F := fun t V => u t V-u t U) (f := fun _ => (0:ℝ)) (fun _ => 2*B)
simp only [integral_zero] at h
apply h
· exact Eventually.of_forall fun t =>
((continuous_haarPolynomialValue _).sub continuous_const).measurable.aestronglyMeasurable
· filter_upwards [eventually_ge_atTop (0:ℝ)] with t ht
exact ae_of_all μ fun V => by
rw [Real.norm_eq_abs]
exact (abs_sub _ _).trans (by linarith [hub t ht V,hub t ht U])
· exact integrable_const _
· exact ae_of_all μ fun V => haarPolynomialHeat_difference_tendsto hN p V U
have he (t : ℝ) : (∫ V, u t V-u t U ∂μ) =
(∫ V, haarPolynomialValue (p : MatrixPolynomial N) V ∂μ)-u t U := by
rw [integral_sub (haarPolynomialValue_integrable _ μ) (integrable_const _),
integral_const,probReal_univ,smul_eq_mul,one_mul]
rw [haarPolynomialHeat_mean]
simp_rw [he] at hlim
have hh : Tendsto (fun t : ℝ => (∫ V, haarPolynomialValue (p : MatrixPolynomial N) V ∂μ)-
((∫ V, haarPolynomialValue (p : MatrixPolynomial N) V ∂μ)-u t U)) atTop
(𝓝 ((∫ V, haarPolynomialValue (p : MatrixPolynomial N) V ∂μ)-0)) :=
tendsto_const_nhds.sub hlim
simpa only [sub_sub_cancel,sub_zero] using hh
end InvariantIsing
end
end OAI
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