| import OAI.Probability.InvariantIsing.Gaussian.GaussianNormMoments |
| import OAI.Probability.InvariantIsing.Spectral.HermitianEigenvalueRoots |
| import OAI.Probability.InvariantIsing.Pressure.RandomSpectralMeasurability |
|
|
| namespace OAI |
|
|
| /-! The actual Gram eigenvalues are bounded by the squared rectangular operator norm. -/ |
| noncomputable section |
| open Matrix |
| open scoped Matrix.Norms.L2Operator |
| namespace InvariantIsing |
|
|
| lemma gaussianPatternCoupling_norm_le {N m : ℕ} (hN : 0 < N) |
| (z : EuclideanSpace ℝ (Fin N × Fin m)) : |
| ‖gaussianPatternCoupling 1 z‖ ≤ (gaussianPatternSingularMax z)^2/N := by |
| have hN' : (0 : ℝ) < N := by exact_mod_cast hN |
| have ht : ‖(gaussianPatternArray z)ᵀ‖ = ‖gaussianPatternArray z‖ := by |
| simpa only [Matrix.conjTranspose_eq_transpose_of_trivial] using |
| Matrix.l2_opNorm_conjTranspose (gaussianPatternArray z) |
| have he : ‖gaussianPatternArray z‖ = gaussianPatternSingularMax z := rfl |
| calc |
| _ = |1/(N : ℝ)| * ‖gaussianPatternArray z*(gaussianPatternArray z)ᵀ‖ := by |
| rw [gaussianPatternCoupling,norm_smul,Real.norm_eq_abs] |
| _ ≤ (1/(N : ℝ))*(‖gaussianPatternArray z‖*‖gaussianPatternArray z‖) := by |
| rw [abs_of_pos (one_div_pos.mpr hN')] |
| exact mul_le_mul_of_nonneg_left ((Matrix.l2_opNorm_mul _ _).trans_eq (by rw [ht])) |
| (by positivity) |
| _ = _ := by rw [he]; ring |
|
|
| lemma gaussianPatternEigenvalues_abs_le {N m : ℕ} (hN : 0 < N) |
| (z : EuclideanSpace ℝ (Fin N × Fin m)) (i : Fin N) : |
| |gaussianPatternEigenvalues z i| ≤ (gaussianPatternSingularMax z)^2/N := by |
| apply le_trans _ (gaussianPatternCoupling_norm_le hN z) |
| unfold gaussianPatternEigenvalues Matrix.IsHermitian.eigenvalues |
| exact hermitian_eigenvalues₀_norm_le _ _ _ |
|
|
| lemma gaussianPattern_spectralRadius_le {n m : ℕ} (c : ℝ) |
| (z : EuclideanSpace ℝ (Fin (n+1) × Fin m)) : |
| spectralRadius (fun i => c*gaussianPatternEigenvalues z i) ≤ |
| |c| * (gaussianPatternSingularMax z)^2/(n+1) := by |
| apply (spectralRadius_le_iff _ _).mpr |
| intro i |
| rw [abs_mul,mul_div_assoc] |
| simpa only [Nat.cast_succ,Nat.cast_add,Nat.cast_one] using |
| mul_le_mul_of_nonneg_left (gaussianPatternEigenvalues_abs_le (Nat.succ_pos n) z i) (abs_nonneg c) |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
|
|
