{"page":"709 - 799","title":"The Altshuler-Shklovskii formulas for random band matrices II: The general case","date_updated":"2025-10-22T10:23:38Z","department":[{"_id":"LaEr"}],"scopus_import":"1","issue":"3","isi":1,"day":"01","abstract":[{"text":"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.\r\n","lang":"eng"}],"publication":"Annales Henri Poincare","publisher":"Springer","language":[{"iso":"eng"}],"oa":1,"project":[{"call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems","grant_number":"338804"}],"date_published":"2015-03-01T00:00:00Z","main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1309.5107"}],"author":[{"orcid":"0000-0001-5366-9603","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","full_name":"Erdös, László","last_name":"Erdös"},{"first_name":"Antti","full_name":"Knowles, Antti","last_name":"Knowles"}],"citation":{"apa":"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>","ama":"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>","chicago":"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>.","ista":"Erdös L, Knowles A. 2015. The Altshuler-Shklovskii formulas for random band matrices II: The general case. Annales Henri Poincare. 16(3), 709–799.","short":"L. Erdös, A. Knowles, Annales Henri Poincare 16 (2015) 709–799.","ieee":"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.","mla":"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>."},"publication_status":"published","intvolume":"        16","date_created":"2018-12-11T11:54:26Z","article_processing_charge":"No","year":"2015","external_id":{"arxiv":["1309.5107"],"isi":["000349364100002"]},"volume":16,"status":"public","arxiv":1,"doi":"10.1007/s00023-014-0333-5","month":"03","ec_funded":1,"publist_id":"5233","_id":"1864","oa_version":"Preprint","type":"journal_article","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"}