| import OAI.Probability.InvariantIsing.Haar.HaarHeatPositivity |
|
|
| namespace OAI |
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|
| /-! Preservation of constants by the constructed polynomial heat flow. -/ |
| noncomputable section |
| open Matrix MvPolynomial |
| open scoped BigOperators |
| namespace InvariantIsing |
|
|
| lemma continuousLinearMap_exp_apply_of_zero {E : Type*} [NormedAddCommGroup E] |
| [NormedSpace ℝ E] [CompleteSpace E] (A : E →L[ℝ] E) (v : E) (hv : A v=0) : |
| NormedSpace.exp A v = v := by |
| have hp (n : ℕ) : (A^(n+1)) v=0 := by |
| induction n with |
| | zero => simpa only [zero_add,pow_one] using hv |
| | succ n ih => |
| rw [show n+1+1=(n+1)+1 by omega,pow_succ',mul_apply_eq_comp,ih,map_zero] |
| have hn (n : ℕ) : ((n.factorial⁻¹ : ℝ) • A^n) v = if n=0 then v else 0 := by |
| cases n with |
| | zero => simp |
| | succ n => simp only [_root_.smul_apply,hp,smul_zero, |
| Nat.succ_ne_zero,ite_false] |
| have hs := (ContinuousLinearMap.apply ℝ E v).hasSum |
| (NormedSpace.exp_series_hasSum_exp' (𝕂 := ℝ) A) |
| have ht : HasSum (fun n : ℕ => if n=0 then v else 0) (NormedSpace.exp A v) := by |
| simpa only [ContinuousLinearMap.apply_apply,hn] using hs |
| exact ht.unique (hasSum_ite_eq 0 v) |
|
|
| lemma haarPolynomialHeat_eq_of_laplacian_zero {N d : ℕ} |
| (p : haarPolynomialSpace N d) (hp : haarPolynomialLaplacian N (p : MatrixPolynomial N)=0) |
| (t : ℝ) : haarPolynomialHeat N d t p = p := by |
| have hr : haarPolynomialRestrictedLaplacian N d p=0 := by |
| apply Subtype.ext |
| exact hp |
| have hc : haarPolynomialCoordinateLaplacian N d (haarPolynomialCoordinates N d p)=0 := by |
| rw [haarPolynomialCoordinateLaplacian_apply,hr,map_zero] |
| apply (haarPolynomialCoordinates N d).injective |
| rw [haarPolynomialCoordinates_heat] |
| apply continuousLinearMap_exp_apply_of_zero |
| simp only [_root_.smul_apply,hc,smul_zero] |
|
|
| lemma haarPolynomialLaplacian_one (N : ℕ) : |
| haarPolynomialLaplacian N (1 : MatrixPolynomial N) = 0 := by |
| simp only [haarPolynomialLaplacian_apply,Derivation.map_one_eq_zero,map_zero, |
| Finset.sum_const_zero] |
|
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| lemma haarPolynomialHeat_one (N d : ℕ) (t : ℝ) : |
| haarPolynomialHeat N d t ⟨1,one_mem_haarPolynomialSpace N d⟩ = |
| ⟨1,one_mem_haarPolynomialSpace N d⟩ := |
| haarPolynomialHeat_eq_of_laplacian_zero _ (haarPolynomialLaplacian_one N) t |
|
|
| end InvariantIsing |
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| end |
|
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| end OAI |
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