| import OAI.Probability.InvariantIsing.Haar.HaarHeatGradient |
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| namespace OAI |
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| /-! A single decaying gradient envelope for each polynomial heat orbit. -/ |
| noncomputable section |
| open Matrix MvPolynomial Filter |
| open scoped Topology |
| namespace InvariantIsing |
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| lemma haarPolynomialHeat_gradient_envelope {N d : ℕ} (p : haarPolynomialSpace N d) : |
| ∃ M : ℝ, 0 ≤ M ∧ ∀ t : ℝ, 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 ≤ |
| (M*Real.exp (-((N:ℝ)-2)*t))^2 := by |
| obtain ⟨B,hB⟩ := haarPolynomialValue_bound (haarPolynomialGamma (p : MatrixPolynomial N) p) |
| let M := max B 1 |
| have hM1 : 1 ≤ M := le_max_right _ _ |
| have hM : 0 ≤ M := by linarith |
| have hC (U : SpecialOrthogonal N) : |
| haarPolynomialValue (haarPolynomialGamma (p : MatrixPolynomial N) p) U ≤ M^2 := by |
| have hb := (le_abs_self _).trans (hB U) |
| have hm : B ≤ M := le_max_left _ _ |
| nlinarith |
| refine ⟨M,hM,?_⟩ |
| intro t ht U |
| have hg := haarPolynomialHeat_gradient_bound p (M^2) hC ht U |
| have he : Real.exp (2*((N:ℝ)-2)*t)*(M*Real.exp (-((N:ℝ)-2)*t))^2 = M^2 := by |
| calc |
| _ = M^2*Real.exp (2*((N:ℝ)-2)*t+ |
| (-((N:ℝ)-2)*t+-((N:ℝ)-2)*t)) := by |
| rw [Real.exp_add,Real.exp_add] |
| ring |
| _ = M^2 := by |
| rw [show 2*((N:ℝ)-2)*t+(-((N:ℝ)-2)*t+-((N:ℝ)-2)*t)=0 by ring, |
| Real.exp_zero,mul_one] |
| have hpos := Real.exp_pos (2*((N:ℝ)-2)*t) |
| nlinarith |
|
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| lemma tendsto_haar_gradient_envelope {N : ℕ} (hN : 3 ≤ N) (M : ℝ) : |
| Tendsto (fun t : ℝ => M*Real.exp (-((N:ℝ)-2)*t)) atTop (𝓝 0) := by |
| have hρ : 0 < (N:ℝ)-2 := by |
| have h3 : (3:ℝ) ≤ N := by exact_mod_cast hN |
| linarith |
| have ht : Tendsto (fun t : ℝ => -((N:ℝ)-2)*t) atTop atBot := by |
| simpa only [Function.comp_def,neg_mul,id_eq] using |
| tendsto_neg_atTop_atBot.comp (tendsto_id.const_mul_atTop hρ) |
| simpa only [Function.comp_def,mul_zero] using (Real.tendsto_exp_atBot.comp ht).const_mul M |
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| end InvariantIsing |
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| end |
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| end OAI |
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