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The M-Wright function PMW[J,\u03c3] with the parameter \u03c3, which used in the main text, is related to M1\/4(x) by the relation PMW[J,\u03c3]=M1\/4(2J\/\u03c3)\/\u03c3, and the latter function was used in Ref.\u00a0[49].","type":"other"},{"key":"4zdv-1ryxCc49R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.132.017101","type":"journal-article"},{"key":"4zdv-1ryxCc44R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.128.160601","type":"journal-article"},{"key":"4zdv-1ryxCc45R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.106.205151","type":"journal-article"},{"key":"4zdv-1ryxCc46R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.109.024417","type":"journal-article"},{"key":"4zdv-1ryxCc47R1","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.2403327121","type":"journal-article"},{"key":"4zdv-1ryxCc48R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.6.013260","type":"journal-article"},{"key":"4zdv-1ryxCc50R1","doi-asserted-by":"publisher","DOI":"10.21468\/SciPostPhys.19.1.027","type":"journal-article"},{"key":"4zdv-1ryxCc51R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.111.024141","type":"journal-article"},{"key":"4zdv-1ryxCc53R1","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511534843","type":"book","volume-title":"The One-Dimensional Hubbard Model","author":"F.\u2009H.\u2009L. Essler","year":"2005"},{"key":"4zdv-1ryxCc54R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevX.6.041065","type":"journal-article"},{"key":"4zdv-1ryxCc55R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.117.207201","type":"journal-article"},{"key":"4zdv-1ryxCc56R1","doi-asserted-by":"publisher","DOI":"10.21468\/SciPostPhysLectNotes.18","type":"journal-article"},{"key":"4zdv-1ryxCc57R1","doi-asserted-by":"publisher","DOI":"10.1088\/1742-5468\/ac257d","type":"journal-article"},{"key":"4zdv-1ryxCc58R1","doi-asserted-by":"publisher","DOI":"10.1088\/1742-5468\/ac3659","type":"journal-article"},{"key":"4zdv-1ryxCc59R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2022.127572","type":"journal-article"},{"key":"4zdv-1ryxCc60R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.131.027101","type":"journal-article"},{"key":"4zdv-1ryxCc61R1","doi-asserted-by":"publisher","DOI":"10.21468\/SciPostPhys.15.4.136","type":"journal-article"},{"key":"4zdv-1ryxCc62R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.134.187101","type":"journal-article"},{"key":"4zdv-1ryxCc64R1","doi-asserted-by":"publisher","DOI":"10.1103\/b587-8yyt","type":"journal-article"},{"key":"4zdv-1ryxCc65R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-021-24726-0","type":"journal-article"},{"key":"4zdv-1ryxCc66R1","doi-asserted-by":"publisher","DOI":"10.21468\/SciPostPhys.15.2.073","type":"journal-article"},{"key":"4zdv-1ryxCc67R1","unstructured":"See Supplemental Material at http:\/\/link.aps.org\/supplemental\/10.1103\/4zdv-1ryx for (I) Derivation of Eq.\u00a0(4), (II) Derivation of Eq.\u00a0(5), (III) Asymptotic analysis of PC[\u0394NR,t], (IV) Asymptotic analysis of PS[\u0394SRz,t], (V) Numerical method for the t\u22120 model, (VI) Slow convergence of PS[0,t], (VII) Asymptotic analysis of the classical automaton, (VIII) Application of ballistic macroscopic fluctuation theory to spinless free fermions, (IX) Discussion on the Fermi-Hubbard model with finite interactions, and (X) Details for the hydrodynamic derivation of the M-Wright function.","type":"other"},{"key":"4zdv-1ryxCc68R1","unstructured":"We comment on the slow convergence around the origin for PS[\u0394SRz,t]. The point of deriving the limiting probability of Eq.\u00a0(7) is to apply the Stirling\u2019s formula to the combination \u200b\u0394NRCn in Eq.\u00a0(5). For \u0394SRz>0, we can safely use this formula. However, for \u0394SRz=0, we need careful calculations and then have additional small terms that do not appear in the case with \u0394SRz>0 for the asymptotic analysis. These small terms lead to the slow convergence. The detailed mathematical explanation is given in Sec.\u00a0VI of SM [67].","type":"other"},{"key":"4zdv-1ryxCc69R1","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.82.1225","type":"journal-article"},{"key":"4zdv-1ryxCc72R1","doi-asserted-by":"publisher","DOI":"10.1016\/S0375-9601(98)00442-3","type":"journal-article"},{"key":"4zdv-1ryxCc73R1","doi-asserted-by":"publisher","DOI":"10.1023\/A:1011324220050","type":"journal-article"},{"key":"4zdv-1ryxCc74R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevX.10.011054","type":"journal-article"},{"key":"4zdv-1ryxCc75R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.125.070602","type":"journal-article"},{"key":"4zdv-1ryxCc76R1","doi-asserted-by":"publisher","DOI":"10.1088\/1742-5468\/ac0f6b","type":"journal-article"}],"container-title":["Physical Review Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.aps.org\/accepted\/10.1103\/4zdv-1ryx","content-type":"application\/pdf","content-version":"am","intended-application":"unspecified"},{"URL":"https:\/\/link.aps.org\/article\/10.1103\/4zdv-1ryx","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/harvest.aps.org\/v2\/journals\/articles\/10.1103\/4zdv-1ryx\/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T18:30:37Z","timestamp":1790879437000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.aps.org\/doi\/10.1103\/4zdv-1ryx"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9,21]]},"references-count":70,"journal-issue":{"issue":"13","published-print":{"date-parts":[[2026,9]]}},"URL":"https:\/\/doi.org\/10.1103\/4zdv-1ryx","relation":{},"ISSN":["0031-9007","1079-7114"],"issn-type":[{"value":"0031-9007","type":"print"},{"value":"1079-7114","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,9,21]]},"assertion":[{"value":"2026-03-12","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"\u00a9 2026 American Physical Society. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright_statement","label":"Copyright statement","group":{"name":"copyright","label":"copyright"}},{"value":"2026","name":"copyright_year","label":"Copyright year","group":{"name":"copyright","label":"copyright"}},{"value":"American Physical Society","name":"copyright_holder","label":"Copyright holder","group":{"name":"copyright","label":"copyright"}}],"article-number":"136301"}}