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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 58 59 60 61 62 | import Mathlib
namespace OAI
/-!
# Differentiating Haar rotation invariance
Infinitesimal Haar invariance gives the integral identity for spectral Ward
identities under explicit domination assumptions.
-/
noncomputable section
open MeasureTheory Filter
open scoped Topology
namespace InvariantIsing
abbrev Orthogonal (N : ℕ) := Matrix.orthogonalGroup (Fin N) ℝ
/-- A dominated infinitesimal left rotation has Haar mean zero. -/
lemma integral_rotation_derivative_eq_zero {N : ℕ} (μ : Measure (Orthogonal N))
[μ.IsMulLeftInvariant] (R : ℝ → Orthogonal N) (hR : R 0 = 1)
(f : Orthogonal N → ℝ) (hf : Measurable f) (hfi : Integrable f μ)
(f' : ℝ → Orthogonal N → ℝ) (hf' : AEStronglyMeasurable (f' 0) μ)
(B : Orthogonal N → ℝ) (hB : Integrable B μ)
(hbound : ∀ᵐ U ∂μ, ∀ t ∈ Set.Ioo (-1 : ℝ) 1, ‖f' t U‖ ≤ B U)
(hderiv : ∀ᵐ U ∂μ, ∀ t ∈ Set.Ioo (-1 : ℝ) 1,
HasDerivAt (fun s => f (R s * U)) (f' t U) t) :
∫ U, f' 0 U ∂μ = 0 := by
have hs : Set.Ioo (-1 : ℝ) 1 ∈ 𝓝 (0 : ℝ) := Ioo_mem_nhds (by norm_num) (by norm_num)
have hm : ∀ᶠ t in 𝓝 (0 : ℝ), AEStronglyMeasurable (fun U => f (R t * U)) μ := by
apply Filter.Eventually.of_forall
intro t
have hmul : Measurable (fun U : Orthogonal N => R t * U) := by
apply Measurable.subtype_mk
change Measurable (fun U : Orthogonal N =>
(R t : Matrix (Fin N) (Fin N) ℝ) * (U : Matrix (Fin N) (Fin N) ℝ))
apply Measurable.of_eval
intro i
apply Measurable.of_eval
intro j
simp only [Matrix.mul_apply]
exact Finset.measurable_sum _ fun k _ =>
measurable_const.mul ((measurable_pi_apply j).comp
((measurable_pi_apply k).comp measurable_subtype_coe))
exact (hf.comp hmul).aestronglyMeasurable
have hi : Integrable (fun U => f (R 0 * U)) μ := by simpa only [hR, one_mul] using hfi
obtain ⟨_, hd⟩ := hasDerivAt_integral_of_dominated_loc_of_deriv_le hs hm hi hf'
hbound hB hderiv
have he : (fun t => ∫ U, f (R t * U) ∂μ) = fun _ : ℝ => ∫ U, f U ∂μ := by
funext t
exact integral_mul_left_eq_self f (R t)
rw [he] at hd
exact hd.unique (hasDerivAt_const (0 : ℝ) (∫ U, f U ∂μ))
end InvariantIsing
end
end OAI
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