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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 | import OAI.Probability.InvariantIsing.Spectral.MeasureRBounds
import OAI.Probability.InvariantIsing.Haar.Rotation
namespace OAI
/-! The degenerate spectral law in the general approximation argument. -/
noncomputable section
open MeasureTheory Filter
open scoped Topology
namespace InvariantIsing
@[simp] lemma measureR_dirac (a x : ℝ) : measureR (Measure.dirac a) a x = a := by
have h := measureR_bounds_all (Measure.dirac a) (l := a) (e := a)
(by simp only [ae_dirac_eq, Filter.eventually_pure, Set.mem_Icc]; exact ⟨le_rfl, le_rfl⟩) x
exact le_antisymm h.2 h.1
@[simp] lemma measureVariational_dirac (a : ℝ) :
variationalFunctional (measureR (Measure.dirac a) a) = ((a / 2 : ℝ) : EReal) := by
have he : measureR (Measure.dirac a) a = fun _ => a := funext (measureR_dirac a)
rw [he, variationalFunctional_constant]
/-- The scalar spectral case of the pressure formula, for arbitrary rotations. -/
theorem tendsto_pressure_scalar_variational
(eig : (N : ℕ) → Fin (N + 1) → ℝ) (U : (N : ℕ) → Rotation (N + 1))
(a : ℝ) (δ : ℕ → ℝ) (hδ : Tendsto δ atTop (𝓝 0))
(heig : ∀ N i, |eig N i - a| ≤ δ N) :
Tendsto (fun N => rotatedPressure (eig N) (U N) (fun _ => 0)) atTop
(𝓝 (variationalFunctional (measureR (Measure.dirac a) a)).toReal) := by
simpa only [measureVariational_dirac, EReal.toReal_coe] using
tendsto_pressure_scalar_spectrum eig U a δ hδ heig
end InvariantIsing
end
end OAI
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