Annales Henri PoincarePub Date : 2025-01-01Epub Date: 2024-12-17DOI: 10.1007/s00023-024-01518-y
László Erdős, Joscha Henheik, Jana Reker, Volodymyr Riabov
{"title":"Prethermalization for Deformed Wigner Matrices.","authors":"László Erdős, Joscha Henheik, Jana Reker, Volodymyr Riabov","doi":"10.1007/s00023-024-01518-y","DOIUrl":"10.1007/s00023-024-01518-y","url":null,"abstract":"<p><p>We prove that a class of weakly perturbed Hamiltonians of the form <math> <mrow><msub><mi>H</mi> <mi>λ</mi></msub> <mo>=</mo> <msub><mi>H</mi> <mn>0</mn></msub> <mo>+</mo> <mi>λ</mi> <mi>W</mi></mrow> </math> , with <i>W</i> being a Wigner matrix, exhibits <i>prethermalization</i>. That is, the time evolution generated by <math><msub><mi>H</mi> <mi>λ</mi></msub> </math> relaxes to its ultimate thermal state via an intermediate prethermal state with a lifetime of order <math><msup><mi>λ</mi> <mrow><mo>-</mo> <mn>2</mn></mrow> </msup> </math> . Moreover, we obtain a general relaxation formula, expressing the perturbed dynamics via the unperturbed dynamics and the ultimate thermal state. The proof relies on a two-resolvent law for the deformed Wigner matrix <math><msub><mi>H</mi> <mi>λ</mi></msub> </math> .</p>","PeriodicalId":72208,"journal":{"name":"Annales Henri Poincare","volume":"26 6","pages":"1991-2033"},"PeriodicalIF":0.0,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12133972/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144236085","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Annales Henri PoincarePub Date : 2025-01-01Epub Date: 2025-02-22DOI: 10.1007/s00023-025-01552-4
Zeév Rudnick, Igor Wigman
{"title":"Almost Sure GOE Fluctuations of Energy Levels for Hyperbolic Surfaces of High Genus.","authors":"Zeév Rudnick, Igor Wigman","doi":"10.1007/s00023-025-01552-4","DOIUrl":"10.1007/s00023-025-01552-4","url":null,"abstract":"<p><p>We study the variance of a linear statistic of the Laplace eigenvalues on a hyperbolic surface, when the surface varies over the moduli space of all surfaces of fixed genus, sampled at random according to the Weil-Petersson measure. The ensemble variance of the linear statistic was recently shown to coincide with that of the corresponding statistic in the Gaussian orthogonal ensemble (GOE) of random matrix theory, in the double limit of first taking large genus and then shrinking size of the energy window. In this note, we show that in this same limit, the (smooth) energy variance for a typical surface is close to the GOE result, a feature called \"ergodicity\" in the random matrix theory literature.</p>","PeriodicalId":72208,"journal":{"name":"Annales Henri Poincare","volume":"26 6","pages":"2279-2291"},"PeriodicalIF":0.0,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12134012/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144236084","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}