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a89d282 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 | 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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