File size: 2,987 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 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 | import OAI.Probability.InvariantIsing.Haar.HaarHeatContinuity
import OAI.Probability.InvariantIsing.Haar.HaarPolynomialGamma
namespace OAI
/-! Joint continuity of differentiated heat polynomials and their squared gradient. -/
noncomputable section
open Matrix MvPolynomial
open scoped BigOperators
namespace InvariantIsing
lemma haarPolynomialOperator_eval_coordinates {N d : ℕ}
(A : Module.End ℝ (MatrixPolynomial N)) (p : haarPolynomialSpace N d)
(M : Matrix (Fin N) (Fin N) ℝ) :
matrixPolynomialEval M (A (p : MatrixPolynomial N)) =
∑ i, haarPolynomialCoordinates N d p i*
matrixPolynomialEval M (A (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_operator_value {N d : ℕ}
(A : Module.End ℝ (MatrixPolynomial N)) (p : haarPolynomialSpace N d) :
Continuous (fun z : ℝ × SpecialOrthogonal N => haarPolynomialValue
(A ((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 (A (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 haarPolynomialOperator_eval_coordinates _ _ _
lemma continuous_haarPolynomialHeat_gamma {N d : ℕ} (p : haarPolynomialSpace N d) :
Continuous (fun z : ℝ × SpecialOrthogonal N => haarPolynomialValue
(haarPolynomialGamma
((haarPolynomialHeat N d z.1 p : haarPolynomialSpace N d) : MatrixPolynomial N)
((haarPolynomialHeat N d z.1 p : haarPolynomialSpace N d) : MatrixPolynomial N)) z.2) := by
have hs : Continuous (fun z : ℝ × SpecialOrthogonal N =>
∑ i : Fin N, ∑ j : Fin N,
(haarPolynomialValue (matrixPolynomialDerivation (planeGenerator i j)
((haarPolynomialHeat N d z.1 p : haarPolynomialSpace N d) : MatrixPolynomial N)) z.2)^2) := by
apply continuous_finsetSum
intro i _
apply continuous_finsetSum
intro j _
exact (continuous_haarPolynomialHeat_operator_value
(matrixPolynomialDerivation (planeGenerator i j)).toLinearMap p).pow 2
convert hs using 1
funext z
simp only [haarPolynomialValue,haarPolynomialGamma,map_sum,map_mul,pow_two]
end InvariantIsing
end
end OAI
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