Asymptotic expansion of 𝜷 matrix models in the multi-cut regime

Article Article scientifique Article Asymptotic expansion of 𝜷 matrix models in the multi-cut regime eng 2024-01-24 2024-01-24 author Borot Gaétan 762697 author Guionnet Alice 10.1017/fms.2023.129 2050-5094 https://www.cambridge.org/core/product/identifier/S2050509423001299/type/journal_article <p>We establish the asymptotic expansion in $\beta $ matrix models with a confining, off-critical potential in the regime where the support of the equilibrium measure is a finite union of segments. We first address the case where the filling fractions of these segments are fixed and show the existence of a $\frac {1}{N}$ expansion. We then study the asymptotics of the sum over the filling fractions to obtain the full asymptotic expansion for the initial problem in the multi-cut regime. In particular, we identify the fluctuations of the linear statistics and show that they are approximated in law by the sum of a Gaussian random variable and an independent Gaussian discrete random variable with oscillating center. Fluctuations of filling fractions are also described by an oscillating discrete Gaussian random variable. We apply our results to study the all-order small dispersion asymptotics of solutions of the Toda chain associated with the one Hermitian matrix model ( $\beta = 2$ ) as well as orthogonal ( $\beta = 1$ ) and skew-orthogonal ( $\beta = 4$ ) polynomials outside the bulk.</p> Dewey 510 25 Section de mathématiques 11 false Forum of mathematics. Sigma 12 e13 25b06656-1386-410e-ad02-4e7ef97a0ef2 Article (Published version) Public Article_scientifique_from_unpaywall.pdf 903326 application/pdf CC BY-4.0 e8fbcbd9aed86a18cb707e78cb45c3fd188e186a1c9bd20ad607493953bb9dac openAccess Faculté des sciences / Section de mathématiques

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Asymptotic expansion of 𝜷 matrix models in the multi-cut regime	Forum of mathematics. Sigma			2024	74	881

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Borot, Gaétan