article published László Erdös author 4DBD5372-F248-11E8-B48F-1D18A9856A870000-0001-5366-9603 Antti Knowles author LaEr department Random matrices, universality and disordered quantum systems project The Altshuler–Shklovskii formulas (Altshuler and Shklovskii, BZh Eksp Teor Fiz 91:200, 1986) predict, for any disordered quantum system in the diffusive regime, a universal power law behaviour for the correlation functions of the mesoscopic eigenvalue density. In this paper and its companion (Erdős and Knowles, The Altshuler–Shklovskii formulas for random band matrices I: the unimodular case, 2013), we prove these formulas for random band matrices. In (Erdős and Knowles, The Altshuler–Shklovskii formulas for random band matrices I: the unimodular case, 2013) we introduced a diagrammatic approach and presented robust estimates on general diagrams under certain simplifying assumptions. In this paper, we remove these assumptions by giving a general estimate of the subleading diagrams. We also give a precise analysis of the leading diagrams which give rise to the Altschuler–Shklovskii power laws. Moreover, we introduce a family of general random band matrices which interpolates between real symmetric (β = 1) and complex Hermitian (β = 2) models, and track the transition for the mesoscopic density–density correlation. Finally, we address the higher-order correlation functions by proving that they behave asymptotically according to a Gaussian process whose covariance is given by the Altshuler–Shklovskii formulas. Springer2015 eng 1309.5107 00034936410000210.1007/s00023-014-0333-5 163709 - 799 L. Erdös and A. Knowles, “The Altshuler-Shklovskii formulas for random band matrices II: The general case,” <i>Annales Henri Poincare</i>, vol. 16, no. 3. Springer, pp. 709–799, 2015. Erdös L, Knowles A. The Altshuler-Shklovskii formulas for random band matrices II: The general case. <i>Annales Henri Poincare</i>. 2015;16(3):709-799. doi:<a href="https://doi.org/10.1007/s00023-014-0333-5">10.1007/s00023-014-0333-5</a> L. Erdös, A. Knowles, Annales Henri Poincare 16 (2015) 709–799. Erdös L, Knowles A. 2015. The Altshuler-Shklovskii formulas for random band matrices II: The general case. Annales Henri Poincare. 16(3), 709–799. Erdös, László, and Antti Knowles. “The Altshuler-Shklovskii Formulas for Random Band Matrices II: The General Case.” <i>Annales Henri Poincare</i>, vol. 16, no. 3, Springer, 2015, pp. 709–99, doi:<a href="https://doi.org/10.1007/s00023-014-0333-5">10.1007/s00023-014-0333-5</a>. Erdös, L., &#38; Knowles, A. (2015). The Altshuler-Shklovskii formulas for random band matrices II: The general case. <i>Annales Henri Poincare</i>. Springer. <a href="https://doi.org/10.1007/s00023-014-0333-5">https://doi.org/10.1007/s00023-014-0333-5</a> Erdös, László, and Antti Knowles. “The Altshuler-Shklovskii Formulas for Random Band Matrices II: The General Case.” <i>Annales Henri Poincare</i>. Springer, 2015. <a href="https://doi.org/10.1007/s00023-014-0333-5">https://doi.org/10.1007/s00023-014-0333-5</a>. 18642018-12-11T11:54:26Z2025-10-22T10:23:38Z