| import OAI.Probability.InvariantIsing.Haar.HaarPolynomialHeat |
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| namespace OAI |
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| /-! Differentiating linear observables of the finite polynomial heat flow. -/ |
| noncomputable section |
| open Matrix MvPolynomial |
| namespace InvariantIsing |
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| def haarPolynomialLinearCoordinates {N d : ℕ} (F : MatrixPolynomial N →ₗ[ℝ] ℝ) : |
| HaarPolynomialCoordinates N d →L[ℝ] ℝ := |
| (F.comp ((haarPolynomialSpace N d).subtype.comp |
| (haarPolynomialCoordinates N d).symm.toLinearMap)).toContinuousLinearMap |
|
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| lemma haarPolynomialLinearCoordinates_apply {N d : ℕ} |
| (F : MatrixPolynomial N →ₗ[ℝ] ℝ) (p : haarPolynomialSpace N d) : |
| haarPolynomialLinearCoordinates F (haarPolynomialCoordinates N d p) = F (p : MatrixPolynomial N) := by |
| change F (((haarPolynomialCoordinates N d).symm (haarPolynomialCoordinates N d p) : |
| haarPolynomialSpace N d) : MatrixPolynomial N) = _ |
| rw [LinearEquiv.symm_apply_apply] |
|
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| lemma haarPolynomialHeat_linear_hasDerivAt {N d : ℕ} |
| (F : MatrixPolynomial N →ₗ[ℝ] ℝ) (p : haarPolynomialSpace N d) (t : ℝ) : |
| HasDerivAt (fun s => F ((haarPolynomialHeat N d s p : haarPolynomialSpace N d) : MatrixPolynomial N)) |
| (F (haarPolynomialLaplacian N |
| ((haarPolynomialHeat N d t p : haarPolynomialSpace N d) : MatrixPolynomial N))) t := by |
| let L := haarPolynomialCoordinateLaplacian N d |
| let v := haarPolynomialCoordinates N d p |
| let ev := (haarPolynomialLinearCoordinates F).comp |
| (ContinuousLinearMap.apply ℝ (HaarPolynomialCoordinates N d) v) |
| have h := ev.hasFDerivAt.comp_hasDerivAt t (hasDerivAt_exp_smul_const' L t) |
| change HasDerivAt (fun s => ev (NormedSpace.exp (s • L))) |
| (ev (L*NormedSpace.exp (t • L))) t at h |
| have he (s : ℝ) : ev (NormedSpace.exp (s • L)) = |
| F ((haarPolynomialHeat N d s p : haarPolynomialSpace N d) : MatrixPolynomial N) := rfl |
| simp_rw [he] at h |
| convert h using 1 |
| change F _ = haarPolynomialLinearCoordinates F (L (NormedSpace.exp (t • L) v)) |
| rw [← haarPolynomialCoordinates_heat,haarPolynomialCoordinateLaplacian_apply] |
| exact (haarPolynomialLinearCoordinates_apply F (haarPolynomialRestrictedLaplacian N d |
| (haarPolynomialHeat N d t p))).symm |
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| end InvariantIsing |
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| end |
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| end OAI |
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