File size: 2,143 Bytes
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 | import OAI.Probability.InvariantIsing.Haar.HaarHeatConstants
namespace OAI
/-! Uniform order bounds for the polynomial heat flow. -/
noncomputable section
open Matrix MvPolynomial Set
namespace InvariantIsing
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⟩
end InvariantIsing
end
end OAI
|