| import OAI.Probability.InvariantIsing.Haar.HaarHeatContinuity |
| import OAI.Probability.InvariantIsing.Haar.HaarPolynomialMinimum |
|
|
| namespace OAI |
|
|
| /-! Positivity of the actual finite polynomial heat flow, by compact comparison. -/ |
| noncomputable section |
| open Matrix MvPolynomial Set |
| open scoped BigOperators |
| namespace InvariantIsing |
|
|
| theorem haarPolynomialHeat_nonneg {N d : ℕ} (p : haarPolynomialSpace N d) |
| (hp : ∀ U : SpecialOrthogonal N, 0 ≤ haarPolynomialValue (p : MatrixPolynomial N) U) |
| {t : ℝ} (ht : 0 ≤ t) (U : SpecialOrthogonal N) : |
| 0 ≤ haarPolynomialValue |
| ((haarPolynomialHeat N d t p : haarPolynomialSpace N d) : MatrixPolynomial N) U := by |
| let q (s : ℝ) : MatrixPolynomial N := Real.exp (-s) • |
| ((haarPolynomialHeat N d s p : haarPolynomialSpace N d) : MatrixPolynomial N) |
| let Wt (z : ℝ × SpecialOrthogonal N) := |
| haarPolynomialValue (haarPolynomialLaplacian N (q z.1)) z.2- |
| haarPolynomialValue (q z.1) z.2 |
| have hq (s : ℝ) (V : SpecialOrthogonal N) : haarPolynomialValue (q s) V = |
| Real.exp (-s)*haarPolynomialValue |
| ((haarPolynomialHeat N d s p : haarPolynomialSpace N d) : MatrixPolynomial N) V := by |
| change matrixPolynomialEval (V : Matrix (Fin N) (Fin N) ℝ) |
| (Real.exp (-s) • _) = _ |
| rw [map_smul] |
| rfl |
| have hL (s : ℝ) (V : SpecialOrthogonal N) : |
| haarPolynomialValue (haarPolynomialLaplacian N (q s)) V = |
| Real.exp (-s)*haarPolynomialValue (haarPolynomialLaplacian N |
| ((haarPolynomialHeat N d s p : haarPolynomialSpace N d) : MatrixPolynomial N)) V := by |
| change matrixPolynomialEval (V : Matrix (Fin N) (Fin N) ℝ) |
| (haarPolynomialLaplacian N (Real.exp (-s) • _)) = _ |
| rw [map_smul,map_smul,smul_eq_mul] |
| rfl |
| have hc : Continuous (fun z : ℝ × SpecialOrthogonal N => haarPolynomialValue (q z.1) z.2) := by |
| simp_rw [hq] |
| exact (Real.continuous_exp.comp continuous_fst.neg).mul (continuous_haarPolynomialHeat_value p) |
| have hi : ∀ V : SpecialOrthogonal N, 0 ≤ haarPolynomialValue (q 0) V := by |
| intro V |
| simpa only [hq,neg_zero,Real.exp_zero,one_mul,haarPolynomialHeat_zero] using hp V |
| have hd (s : ℝ) (_ : s ∈ Ioc (0 : ℝ) t) (V : SpecialOrthogonal N) : |
| HasDerivAt (fun r => haarPolynomialValue (q r) V) (Wt (s,V)) s := by |
| have he : HasDerivAt (fun r : ℝ => Real.exp (-r)) (-Real.exp (-s)) s := by |
| convert ((hasDerivAt_id s).fun_neg).exp using 1 |
| all_goals simp only [id_eq,mul_neg,mul_one] |
| have hv := haarPolynomialHeat_hasDerivAt p (V : Matrix (Fin N) (Fin N) ℝ) s |
| convert he.mul hv using 1 |
| · funext r |
| exact hq r V |
| · change haarPolynomialValue (haarPolynomialLaplacian N (q s)) V- |
| haarPolynomialValue (q s) V = _ |
| rw [hL,hq] |
| simp only [haarPolynomialValue] |
| ring |
| have hz := haarPolynomial_parabolic_nonneg ht q Wt hc.continuousOn hi hd |
| (fun _ _ _ => le_rfl) t ⟨ht,le_rfl⟩ U |
| rw [hq] at hz |
| exact nonneg_of_mul_nonneg_right hz (Real.exp_pos (-t)) |
|
|
| end InvariantIsing |
|
|
| end |
|
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| end OAI |
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