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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 84 85 86 87 88 89 90 91 92 93 94 95 96 97 | import OAI.Probability.InvariantIsing.Haar.HaarHeatLogBarrier
import OAI.Probability.InvariantIsing.Haar.HaarHeatOrder
import OAI.Probability.InvariantIsing.Haar.CompactNegativeMinimum
namespace OAI
/-! Exponential decay of the logarithmic gradient of a positive polynomial heat orbit. -/
noncomputable section
open Matrix MvPolynomial Set
namespace InvariantIsing
theorem haarPolynomialHeat_log_gradient_bound {N d : ℕ} (p : haarPolynomialSpace N d)
(ε C : ℝ) (hε : 0 < ε)
(hp : ∀ U : SpecialOrthogonal N, ε ≤ haarPolynomialValue (p : MatrixPolynomial N) U)
(hC : ∀ U : SpecialOrthogonal N,
haarPolynomialValue (haarPolynomialGamma (p : MatrixPolynomial N) p) U ≤
C*(haarPolynomialValue (p : MatrixPolynomial N) U)^2)
{T : ℝ} (hT : 0 ≤ T) (U : SpecialOrthogonal N) :
haarPolynomialValue (haarPolynomialGamma
((haarPolynomialHeat N d T p : haarPolynomialSpace N d) : MatrixPolynomial N)
((haarPolynomialHeat N d T p : haarPolynomialSpace N d) : MatrixPolynomial N)) U ≤
C*Real.exp (-2*((N:ℝ)-2)*T)*
(haarPolynomialValue ((haarPolynomialHeat N d T p : haarPolynomialSpace N d) : MatrixPolynomial N) U)^2 := by
let q (s : ℝ) : MatrixPolynomial N := (haarPolynomialHeat N d s p : haarPolynomialSpace N d)
let a (s : ℝ) := C*Real.exp (-2*((N:ℝ)-2)*s)
let H (s : ℝ) (V : SpecialOrthogonal N) := haarPolynomialValue (haarHeatLogBarrier p C s) V
let G (s : ℝ) (V : SpecialOrthogonal N) := haarPolynomialValue (haarPolynomialGamma (q s) (q s)) V
obtain ⟨P,hP⟩ := haarPolynomialValue_bound (p : MatrixPolynomial N)
have hlow (s : ℝ) (hs : 0 ≤ s) (V : SpecialOrthogonal N) : ε ≤ haarPolynomialValue (q s) V :=
(haarPolynomialHeat_bounds p ε P hp (fun W => (le_abs_self _).trans (hP W)) hs V).1
obtain ⟨B,hB⟩ := (isCompact_Icc.prod (isCompact_univ : IsCompact (Set.univ : Set (SpecialOrthogonal N)))).exists_bound_of_continuousOn
(continuous_haarPolynomialHeat_gamma p).continuousOn
have hB0 : 0 ≤ B := (norm_nonneg _).trans (hB (0,1) ⟨⟨le_rfl,hT⟩,mem_univ _⟩)
have hGB (s : ℝ) (hs : s ∈ Icc (0:ℝ) T) (V : SpecialOrthogonal N) : G s V ≤ B :=
(le_abs_self _).trans (hB (s,V) ⟨hs,mem_univ _⟩)
let A := 2*|((N:ℝ)-2)|+2*B/ε^2+1
let W (s : ℝ) (V : SpecialOrthogonal N) := Real.exp (-A*s)*H s V
let D (s : ℝ) (V : SpecialOrthogonal N) :=
-2*((N:ℝ)-2)*a s*(haarPolynomialValue (q s) V)^2+
2*a s*haarPolynomialValue (q s) V*haarPolynomialValue (haarPolynomialLaplacian N (q s)) V-
2*haarPolynomialValue (haarPolynomialGamma (q s) (haarPolynomialLaplacian N (q s))) V
let Wt (z : ℝ × SpecialOrthogonal N) := Real.exp (-A*z.1)*(D z.1 z.2-A*H z.1 z.2)
have hc : Continuous (fun z : ℝ × SpecialOrthogonal N => W z.1 z.2) :=
(Real.continuous_exp.comp (continuous_const.mul continuous_fst)).mul
(continuous_haarHeatLogBarrier p C)
have hi (V : SpecialOrthogonal N) : 0 ≤ W 0 V := by
simp only [W,H,haarHeatLogBarrier,haarPolynomialHeat_zero,mul_zero,Real.exp_zero,mul_one,one_mul]
change 0 ≤ matrixPolynomialEval (V : Matrix (Fin N) (Fin N) ℝ) (haarLogBarrier p C)
rw [haarLogBarrier_eval]
exact sub_nonneg.mpr (hC V)
have hd (s : ℝ) (_ : s ∈ Ioc (0:ℝ) T) (V : SpecialOrthogonal N) :
HasDerivAt (fun r => W r V) (Wt (s,V)) s := by
have he : HasDerivAt (fun r => Real.exp (-A*r)) (-A*Real.exp (-A*s)) s := by
convert ((hasDerivAt_id s).const_mul (-A)).exp using 1
all_goals simp only [id_eq,mul_one]
all_goals ring
have hh := haarHeatLogBarrier_hasDerivAt p C s V
convert he.mul hh using 1
all_goals dsimp only [Wt,D,H,q,a]
all_goals ring
have hmin (s : ℝ) (hs : s ∈ Ioc (0:ℝ) T) (V : SpecialOrthogonal N)
(hneg : W s V < 0) (hm : ∀ Z, W s V ≤ W s Z) : 0 < Wt (s,V) := by
have hHneg : H s V < 0 := by
have he := Real.exp_pos (-A*s)
dsimp only [W] at hneg
nlinarith
have hHmin (Z : SpecialOrthogonal N) : H s V ≤ H s Z :=
(mul_le_mul_iff_right₀ (Real.exp_pos (-A*s))).mp (hm Z)
have hu : 0 < haarPolynomialValue (q s) V := hε.trans_le (hlow s hs.1.le V)
have hD : (-2*((N:ℝ)-2)+2*G s V/(haarPolynomialValue (q s) V)^2)*H s V ≤ D s V :=
haarLogBarrier_min_derivative_lower (q s) (a s) V hu.ne' hHmin
have hw : G s V/(haarPolynomialValue (q s) V)^2 ≤ B/ε^2 := by
calc
_ ≤ B/(haarPolynomialValue (q s) V)^2 :=
div_le_div_of_nonneg_right (hGB s ⟨hs.1.le,hs.2⟩ V) (sq_nonneg _)
_ ≤ B/ε^2 := div_le_div_of_nonneg_left hB0 (sq_pos_of_pos hε)
((sq_le_sq₀ hε.le hu.le).mpr (hlow s hs.1.le V))
have hcoef : -2*((N:ℝ)-2)+2*G s V/(haarPolynomialValue (q s) V)^2 ≤ A-1 := by
dsimp only [A]
linear_combination 2*hw+2*(neg_le_abs ((N:ℝ)-2))
have hmulf := mul_le_mul_of_nonpos_right hcoef hHneg.le
have hstrict : 0 < D s V-A*H s V := by nlinarith
exact mul_pos (Real.exp_pos (-A*s)) hstrict
have hz := compact_nonneg_of_negative_min_derivative hT W Wt hc.continuousOn hi hd hmin
T ⟨hT,le_rfl⟩ U
have hH : 0 ≤ H T U := (mul_nonneg_iff_of_pos_left (Real.exp_pos (-A*T))).mp hz
change 0 ≤ matrixPolynomialEval (U : Matrix (Fin N) (Fin N) ℝ)
(haarLogBarrier (q T) (a T)) at hH
rw [haarLogBarrier_eval] at hH
exact sub_nonneg.mp hH
end InvariantIsing
end
end OAI
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