| import OAI.Probability.InvariantIsing.Haar.HaarHeatMixing |
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| namespace OAI |
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| /-! Continuous tests of the polynomial heat flow and their Haar limits. -/ |
| noncomputable section |
| open Matrix MvPolynomial MeasureTheory Filter Set |
| open scoped Topology |
| namespace InvariantIsing |
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| lemma haarPolynomialHeat_test_bound {N d : ℕ} (p : haarPolynomialSpace N d) |
| (F : ℝ → ℝ) (hF : Continuous F) : |
| ∃ B : ℝ, ∀ t : ℝ, 0 ≤ t → ∀ U : SpecialOrthogonal N, |
| ‖F (haarPolynomialValue |
| ((haarPolynomialHeat N d t p : haarPolynomialSpace N d) : MatrixPolynomial N) U)‖ ≤ B := by |
| obtain ⟨P,hP⟩ := haarPolynomialValue_bound (p : MatrixPolynomial N) |
| obtain ⟨B,hB⟩ := isCompact_Icc.exists_bound_of_continuousOn hF.continuousOn |
| refine ⟨B,?_⟩ |
| intro t ht U |
| apply hB |
| exact haarPolynomialHeat_bounds p (-P) P |
| (fun V => (abs_le.mp (hP V)).1) (fun V => (abs_le.mp (hP V)).2) ht U |
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| theorem haarPolynomialHeat_test_tendsto_mean {N d : ℕ} (hN : 3 ≤ N) |
| (μ : Measure (SpecialOrthogonal N)) [IsProbabilityMeasure μ] [μ.IsMulLeftInvariant] |
| (p : haarPolynomialSpace N d) (F : ℝ → ℝ) (hF : Continuous F) : |
| Tendsto (fun t : ℝ => ∫ U, F (haarPolynomialValue |
| ((haarPolynomialHeat N d t p : haarPolynomialSpace N d) : MatrixPolynomial N) U) ∂μ) |
| atTop (𝓝 (F (∫ U, haarPolynomialValue (p : MatrixPolynomial N) U ∂μ))) := by |
| obtain ⟨B,hB⟩ := haarPolynomialHeat_test_bound p F hF |
| have h := tendsto_integral_filter_of_dominated_convergence (μ := μ) (l := atTop) |
| (F := fun t U => F (haarPolynomialValue |
| ((haarPolynomialHeat N d t p : haarPolynomialSpace N d) : MatrixPolynomial N) U)) |
| (f := fun _ => F (∫ U, haarPolynomialValue (p : MatrixPolynomial N) U ∂μ)) (fun _ => B) |
| simp only [integral_const,probReal_univ,smul_eq_mul,one_mul] at h |
| apply h |
| · exact Eventually.of_forall fun t => |
| (hF.comp (continuous_haarPolynomialValue _)).measurable.aestronglyMeasurable |
| · filter_upwards [eventually_ge_atTop (0:ℝ)] with t ht |
| exact ae_of_all μ (hB t ht) |
| · exact integrable_const _ |
| · exact ae_of_all μ fun U => hF.continuousAt.tendsto.comp (haarPolynomialHeat_tendsto_mean hN μ p U) |
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| theorem continuousOn_haarPolynomialHeat_test_integral {N d : ℕ} |
| (μ : Measure (SpecialOrthogonal N)) [IsFiniteMeasure μ] |
| (p : haarPolynomialSpace N d) (F : ℝ → ℝ) (hF : Continuous F) : |
| ContinuousOn (fun t : ℝ => ∫ U, F (haarPolynomialValue |
| ((haarPolynomialHeat N d t p : haarPolynomialSpace N d) : MatrixPolynomial N) U) ∂μ) (Ici 0) := by |
| obtain ⟨B,hB⟩ := haarPolynomialHeat_test_bound p F hF |
| intro t _ |
| apply tendsto_integral_filter_of_dominated_convergence (fun _ => B) |
| · exact Eventually.of_forall fun s => |
| (hF.comp (continuous_haarPolynomialValue _)).measurable.aestronglyMeasurable |
| · filter_upwards [self_mem_nhdsWithin] with s hs |
| exact ae_of_all μ (hB s hs) |
| · exact integrable_const _ |
| · exact ae_of_all μ fun U => ((hF.comp |
| ((continuous_haarPolynomialHeat_value p).comp (continuous_id.prodMk continuous_const))).continuousAt).tendsto.mono_left |
| nhdsWithin_le_nhds |
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| end InvariantIsing |
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| end |
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| end OAI |
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