| import OAI.Probability.InvariantIsing.Gaussian.GaussianEmpiricalWeak | |
| namespace OAI | |
| /-! Continuity of finite empirical spectral measures in their actual eigenvalues. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| open scoped Topology | |
| namespace InvariantIsing | |
| theorem continuous_empiricalSpectralLaw {N : ℕ} (hN : 0 < N) : | |
| Continuous (fun eig : Fin N → ℝ => empiricalSpectralLaw hN eig) := by | |
| apply ProbabilityMeasure.continuous_iff_forall_continuous_integral.mpr | |
| intro f | |
| simp only [empiricalSpectralLaw_integral] | |
| exact continuous_const.mul (continuous_finsetSum _ fun i _ => f.continuous.comp (continuous_apply i)) | |
| theorem continuous_scaled_gaussianEmpirical {N m : ℕ} (hN : 0 < N) (c : ℝ) : | |
| Continuous (fun z : EuclideanSpace ℝ (Fin N × Fin m) => LevyProkhorov.ofMeasure | |
| (empiricalSpectralLaw hN (fun i => c*gaussianPatternEigenvalues z i))) := by | |
| apply LevyProkhorov.continuous_ofMeasure_probabilityMeasure.comp | |
| apply (continuous_empiricalSpectralLaw hN).comp | |
| apply continuous_pi | |
| intro i | |
| exact continuous_const.mul ((continuous_apply i).comp (continuous_gaussianPatternEigenvalues N m)) | |
| end InvariantIsing | |
| end | |
| end OAI | |