| import OAI.Probability.InvariantIsing.Haar.HaarPolynomialHeat |
|
|
| namespace OAI |
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| /-! Joint continuity of the finite polynomial heat flow. -/ |
| noncomputable section |
| open Matrix MvPolynomial |
| open scoped BigOperators |
| namespace InvariantIsing |
|
|
| def haarPolynomialBasis (N d : ℕ) |
| (i : Fin (Module.finrank ℝ (haarPolynomialSpace N d))) : MatrixPolynomial N := |
| ((Module.finBasis ℝ (haarPolynomialSpace N d)) i : MatrixPolynomial N) |
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| lemma haarPolynomialEval_eq_coordinates {N d : ℕ} |
| (p : haarPolynomialSpace N d) (M : Matrix (Fin N) (Fin N) ℝ) : |
| matrixPolynomialEval M (p : MatrixPolynomial N) = |
| ∑ i, haarPolynomialCoordinates N d p i* |
| matrixPolynomialEval M (haarPolynomialBasis N d i) := by |
| have hp : p = ∑ i, haarPolynomialCoordinates N d p i • |
| (Module.finBasis ℝ (haarPolynomialSpace N d)) i := by |
| simpa only [haarPolynomialCoordinates,LinearEquiv.symm_apply_apply] using |
| ((Module.finBasis ℝ (haarPolynomialSpace N d)).equivFun_symm_apply |
| (haarPolynomialCoordinates N d p)) |
| conv_lhs => rw [hp] |
| simp only [Submodule.coe_sum,Submodule.coe_smul,map_sum,map_smul,smul_eq_mul, |
| haarPolynomialBasis] |
|
|
| lemma continuous_haarPolynomialHeat_coordinates {N d : ℕ} (p : haarPolynomialSpace N d) : |
| Continuous (fun t : ℝ => haarPolynomialCoordinates N d (haarPolynomialHeat N d t p)) := by |
| let L := haarPolynomialCoordinateLaplacian N d |
| have hc : Continuous (fun t : ℝ => NormedSpace.exp (t • L)) := |
| continuous_iff_continuousAt.mpr fun t => (hasDerivAt_exp_smul_const' L t).continuousAt |
| have hh := hc.clm_apply (continuous_const (y := haarPolynomialCoordinates N d p)) |
| simpa only [haarPolynomialCoordinates_heat] using hh |
|
|
| lemma continuous_haarPolynomialHeat_value {N d : ℕ} (p : haarPolynomialSpace N d) : |
| Continuous (fun z : ℝ × SpecialOrthogonal N => haarPolynomialValue |
| ((haarPolynomialHeat N d z.1 p : haarPolynomialSpace N d) : MatrixPolynomial N) z.2) := by |
| have hc := continuous_haarPolynomialHeat_coordinates p |
| have hs : Continuous (fun z : ℝ × SpecialOrthogonal N => |
| ∑ i, haarPolynomialCoordinates N d (haarPolynomialHeat N d z.1 p) i* |
| haarPolynomialValue (haarPolynomialBasis N d i) z.2) := by |
| apply continuous_finsetSum |
| intro i _ |
| exact (((continuous_apply i).comp hc).comp continuous_fst).mul |
| ((continuous_haarPolynomialValue _).comp continuous_snd) |
| convert hs using 1 |
| funext z |
| exact haarPolynomialEval_eq_coordinates _ _ |
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|
| end InvariantIsing |
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| end |
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| end OAI |
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