| import OAI.Probability.InvariantIsing.Haar.HaarHeatConstants |
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| namespace OAI |
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| /-! Uniform order bounds for the polynomial heat flow. -/ |
| noncomputable section |
| open Matrix MvPolynomial Set |
| namespace InvariantIsing |
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| theorem haarPolynomialHeat_bounds {N d : ℕ} (p : haarPolynomialSpace N d) |
| (a b : ℝ) |
| (ha : ∀ U : SpecialOrthogonal N, a ≤ haarPolynomialValue (p : MatrixPolynomial N) U) |
| (hb : ∀ U : SpecialOrthogonal N, haarPolynomialValue (p : MatrixPolynomial N) U ≤ b) |
| {t : ℝ} (ht : 0 ≤ t) (U : SpecialOrthogonal N) : |
| haarPolynomialValue |
| ((haarPolynomialHeat N d t p : haarPolynomialSpace N d) : MatrixPolynomial N) U ∈ Icc a b := by |
| let onep : haarPolynomialSpace N d := ⟨1,one_mem_haarPolynomialSpace N d⟩ |
| have h1 : haarPolynomialHeat N d t onep = onep := haarPolynomialHeat_one N d t |
| have hscalar (c : ℝ) (V : SpecialOrthogonal N) : |
| matrixPolynomialEval (V : Matrix (Fin N) (Fin N) ℝ) |
| ((c • onep : haarPolynomialSpace N d) : MatrixPolynomial N) = c := by |
| change matrixPolynomialEval (V : Matrix (Fin N) (Fin N) ℝ) |
| (c • (1 : MatrixPolynomial N)) = c |
| rw [map_smul,map_one] |
| exact mul_one c |
| have hl : ∀ V : SpecialOrthogonal N, |
| 0 ≤ haarPolynomialValue ((p-a • onep : haarPolynomialSpace N d) : MatrixPolynomial N) V := by |
| intro V |
| change 0 ≤ matrixPolynomialEval (V : Matrix (Fin N) (Fin N) ℝ) (_-_) |
| rw [map_sub,hscalar] |
| exact sub_nonneg.mpr (ha V) |
| have hu : ∀ V : SpecialOrthogonal N, |
| 0 ≤ haarPolynomialValue ((b • onep-p : haarPolynomialSpace N d) : MatrixPolynomial N) V := by |
| intro V |
| change 0 ≤ matrixPolynomialEval (V : Matrix (Fin N) (Fin N) ℝ) (_-_) |
| rw [map_sub,hscalar] |
| exact sub_nonneg.mpr (hb V) |
| have hlow := haarPolynomialHeat_nonneg (p-a • onep) hl ht U |
| have hupp := haarPolynomialHeat_nonneg (b • onep-p) hu ht U |
| rw [map_sub,map_smul,h1] at hlow hupp |
| change 0 ≤ matrixPolynomialEval (U : Matrix (Fin N) (Fin N) ℝ) (_-_) at hlow hupp |
| rw [map_sub,hscalar] at hlow hupp |
| exact ⟨sub_nonneg.mp hlow,sub_nonneg.mp hupp⟩ |
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| end InvariantIsing |
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| end |
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| end OAI |
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