| 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 |
|
|
