Asymptotic expansion of $\beta $ matrix models in the multi-cut regime
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 exi...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2024-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509423001299/type/journal_article |
| _version_ | 1797347680604651520 |
|---|---|
| author | Gaëtan Borot Alice Guionnet |
| author_facet | Gaëtan Borot Alice Guionnet |
| author_sort | Gaëtan Borot |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-08T11:52:16Z |
| format | Article |
| id | doaj.art-560e63b727764478adfdfeb1c9b22a73 |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-03-08T11:52:16Z |
| publishDate | 2024-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-560e63b727764478adfdfeb1c9b22a732024-01-24T08:22:06ZengCambridge University PressForum of Mathematics, Sigma2050-50942024-01-011210.1017/fms.2023.129Asymptotic expansion of $\beta $ matrix models in the multi-cut regimeGaëtan Borot0Alice Guionnet1The work has been conducted at Section de Mathématiques, Université de Genève, at MIT, Department of Mathematics, at MPIM Bonn, and at (current address) Humboldt-Universität zu Berlin, Institut für Mathematik und Institut für Physik, Unter den Linden 6, Berlin 10099, Germany; E-mail:The work has been conducted at MIT, Department of Mathematics and (current address) UMPA, CNRS UMR 5669, ENS Lyon, 46 allée d’Italie, Lyon 69007, France; E-mail: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.https://www.cambridge.org/core/product/identifier/S2050509423001299/type/journal_article60B2015B5260F05 |
| spellingShingle | Gaëtan Borot Alice Guionnet Asymptotic expansion of $\beta $ matrix models in the multi-cut regime Forum of Mathematics, Sigma 60B20 15B52 60F05 |
| title | Asymptotic expansion of $\beta $ matrix models in the multi-cut regime |
| title_full | Asymptotic expansion of $\beta $ matrix models in the multi-cut regime |
| title_fullStr | Asymptotic expansion of $\beta $ matrix models in the multi-cut regime |
| title_full_unstemmed | Asymptotic expansion of $\beta $ matrix models in the multi-cut regime |
| title_short | Asymptotic expansion of $\beta $ matrix models in the multi-cut regime |
| title_sort | asymptotic expansion of beta matrix models in the multi cut regime |
| topic | 60B20 15B52 60F05 |
| url | https://www.cambridge.org/core/product/identifier/S2050509423001299/type/journal_article |
| work_keys_str_mv | AT gaetanborot asymptoticexpansionofbetamatrixmodelsinthemulticutregime AT aliceguionnet asymptoticexpansionofbetamatrixmodelsinthemulticutregime |