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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 | import OAI.Probability.InvariantIsing.Haar.HaarPolynomialHeat
namespace OAI
/-! Joint continuity of the finite polynomial heat flow. -/
noncomputable section
open Matrix MvPolynomial
open scoped BigOperators
namespace InvariantIsing
def haarPolynomialBasis (N d : ℕ)
(i : Fin (Module.finrank ℝ (haarPolynomialSpace N d))) : MatrixPolynomial N :=
((Module.finBasis ℝ (haarPolynomialSpace N d)) i : MatrixPolynomial N)
lemma haarPolynomialEval_eq_coordinates {N d : ℕ}
(p : haarPolynomialSpace N d) (M : Matrix (Fin N) (Fin N) ℝ) :
matrixPolynomialEval M (p : MatrixPolynomial N) =
∑ i, haarPolynomialCoordinates N d p i*
matrixPolynomialEval M (haarPolynomialBasis N d i) := by
have hp : p = ∑ i, haarPolynomialCoordinates N d p i •
(Module.finBasis ℝ (haarPolynomialSpace N d)) i := by
simpa only [haarPolynomialCoordinates,LinearEquiv.symm_apply_apply] using
((Module.finBasis ℝ (haarPolynomialSpace N d)).equivFun_symm_apply
(haarPolynomialCoordinates N d p))
conv_lhs => rw [hp]
simp only [Submodule.coe_sum,Submodule.coe_smul,map_sum,map_smul,smul_eq_mul,
haarPolynomialBasis]
lemma continuous_haarPolynomialHeat_coordinates {N d : ℕ} (p : haarPolynomialSpace N d) :
Continuous (fun t : ℝ => haarPolynomialCoordinates N d (haarPolynomialHeat N d t p)) := by
let L := haarPolynomialCoordinateLaplacian N d
have hc : Continuous (fun t : ℝ => NormedSpace.exp (t • L)) :=
continuous_iff_continuousAt.mpr fun t => (hasDerivAt_exp_smul_const' L t).continuousAt
have hh := hc.clm_apply (continuous_const (y := haarPolynomialCoordinates N d p))
simpa only [haarPolynomialCoordinates_heat] using hh
lemma continuous_haarPolynomialHeat_value {N d : ℕ} (p : haarPolynomialSpace N d) :
Continuous (fun z : ℝ × SpecialOrthogonal N => haarPolynomialValue
((haarPolynomialHeat N d z.1 p : haarPolynomialSpace N d) : MatrixPolynomial N) z.2) := by
have hc := continuous_haarPolynomialHeat_coordinates p
have hs : Continuous (fun z : ℝ × SpecialOrthogonal N =>
∑ i, haarPolynomialCoordinates N d (haarPolynomialHeat N d z.1 p) i*
haarPolynomialValue (haarPolynomialBasis N d i) z.2) := by
apply continuous_finsetSum
intro i _
exact (((continuous_apply i).comp hc).comp continuous_fst).mul
((continuous_haarPolynomialValue _).comp continuous_snd)
convert hs using 1
funext z
exact haarPolynomialEval_eq_coordinates _ _
end InvariantIsing
end
end OAI
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