| import OAI.Probability.InvariantIsing.Haar.HaarHeatGammaDerivative |
| import OAI.Probability.InvariantIsing.Haar.HaarHeatOperator |
| import OAI.Probability.InvariantIsing.Haar.HaarPolynomialEvolution |
| import OAI.Probability.InvariantIsing.Haar.HaarPolynomialCurvature |
| import OAI.Probability.InvariantIsing.Haar.HaarHeatConstants |
|
|
| namespace OAI |
|
|
| /-! Exponential squared-gradient decay for the actual orthogonal polynomial heat flow. -/ |
| noncomputable section |
| open Matrix MvPolynomial Set |
| namespace InvariantIsing |
|
|
| theorem haarPolynomialHeat_gradient_bound {N d : ℕ} |
| (p : haarPolynomialSpace N d) (C : ℝ) |
| (hC : ∀ U : SpecialOrthogonal N, |
| haarPolynomialValue (haarPolynomialGamma (p : MatrixPolynomial N) p) U ≤ C) |
| {t : ℝ} (ht : 0 ≤ t) (U : SpecialOrthogonal N) : |
| Real.exp (2*((N : ℝ)-2)*t)*haarPolynomialValue (haarPolynomialGamma |
| ((haarPolynomialHeat N d t p : haarPolynomialSpace N d) : MatrixPolynomial N) |
| ((haarPolynomialHeat N d t p : haarPolynomialSpace N d) : MatrixPolynomial N)) U ≤ C := by |
| let u (s : ℝ) : MatrixPolynomial N := (haarPolynomialHeat N d s p : haarPolynomialSpace N d) |
| let g (s : ℝ) := haarPolynomialGamma (u s) (u s) |
| let k : ℝ := 2*((N : ℝ)-2) |
| let q (s : ℝ) : MatrixPolynomial N := C • 1-Real.exp (k*s) • g s |
| let dq (s : ℝ) (V : SpecialOrthogonal N) := |
| -Real.exp (k*s)*(k*haarPolynomialValue (g s) V+ |
| 2*haarPolynomialValue (haarPolynomialGamma (u s) (haarPolynomialLaplacian N (u s))) V) |
| have heval (s : ℝ) (V : SpecialOrthogonal N) : |
| haarPolynomialValue (q s) V = C-Real.exp (k*s)*haarPolynomialValue (g s) V := by |
| change matrixPolynomialEval (V : Matrix (Fin N) (Fin N) ℝ) (_-_) = _ |
| simp only [map_sub,map_smul,map_one,smul_eq_mul,mul_one,haarPolynomialValue] |
| have hL (s : ℝ) (V : SpecialOrthogonal N) : |
| haarPolynomialValue (haarPolynomialLaplacian N (q s)) V = |
| -Real.exp (k*s)*haarPolynomialValue (haarPolynomialLaplacian N (g s)) V := by |
| change matrixPolynomialEval (V : Matrix (Fin N) (Fin N) ℝ) |
| (haarPolynomialLaplacian N (C • 1-Real.exp (k*s) • g s)) = _ |
| rw [map_sub,map_smul,map_smul,haarPolynomialLaplacian_one,smul_zero,zero_sub, |
| map_neg,map_smul,smul_eq_mul] |
| simp only [haarPolynomialValue,neg_mul] |
| have hc : Continuous (fun z : ℝ × SpecialOrthogonal N => haarPolynomialValue (q z.1) z.2) := by |
| simp_rw [heval] |
| exact continuous_const.sub |
| ((Real.continuous_exp.comp (continuous_const.mul continuous_fst)).mul |
| (continuous_haarPolynomialHeat_gamma p)) |
| have hi : ∀ V : SpecialOrthogonal N, 0 ≤ haarPolynomialValue (q 0) V := by |
| intro V |
| rw [heval] |
| simpa only [g,u,haarPolynomialHeat_zero,mul_zero,Real.exp_zero,one_mul] using |
| sub_nonneg.mpr (hC V) |
| have hd (s : ℝ) (_ : s ∈ Ioc (0 : ℝ) t) (V : SpecialOrthogonal N) : |
| HasDerivAt (fun r => haarPolynomialValue (q r) V) (dq s V) s := by |
| have he : HasDerivAt (fun r => Real.exp (k*r)) (k*Real.exp (k*s)) s := by |
| convert ((hasDerivAt_id s).const_mul k).exp using 1 |
| all_goals simp only [id_eq] |
| all_goals ring |
| have hg := haarPolynomialHeat_gamma_hasDerivAt p (V : Matrix (Fin N) (Fin N) ℝ) s |
| convert (hasDerivAt_const s C).sub (he.mul hg) using 1 |
| · funext r |
| exact heval r V |
| · dsimp only [dq,haarPolynomialValue,g,u] |
| ring |
| have hpde (s : ℝ) (_ : s ∈ Ioc (0 : ℝ) t) (V : SpecialOrthogonal N) : |
| haarPolynomialValue (haarPolynomialLaplacian N (q s)) V ≤ dq s V := by |
| have hb := matrixPolynomial_curvature (u s) (V : Matrix (Fin N) (Fin N) ℝ) |
| have hh : k*haarPolynomialValue (g s) V+ |
| 2*haarPolynomialValue (haarPolynomialGamma (u s) (haarPolynomialLaplacian N (u s))) V ≤ |
| haarPolynomialValue (haarPolynomialLaplacian N (g s)) V := by |
| dsimp only [k,g,haarPolynomialValue] |
| linarith |
| rw [hL] |
| dsimp only [dq] |
| nlinarith [mul_le_mul_of_nonneg_left hh (Real.exp_pos (k*s)).le] |
| have hz := haarPolynomial_evolution_nonneg ht q dq hc.continuousOn hi hd hpde t ⟨ht,le_rfl⟩ U |
| rw [heval] at hz |
| exact sub_nonneg.mp hz |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
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|
