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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Large deviation principle for some beta ensembles
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by Tien-Cuong Dinh and Viêt-Anh Nguyên PDF
Trans. Amer. Math. Soc. 370 (2018), 6565-6584 Request permission


Let $L$ be a positive line bundle over a projective complex manifold $X$, $L^p$ its tensor power of order $p$, $H^0(X,L^p)$ the space of holomorphic sections of $L^p$, and $N_p$ the complex dimension of $H^0(X,L^p)$. The determinant of a basis of $H^0(X,L^p)$, together with some given probability measure on a weighted compact set in $X$, induces naturally a $\beta$-ensemble, i.e., a random $N_p$-point process on the compact set. Physically, depending on $X$ and the value of $\beta$, this general setting corresponds to a gas of free or interacting fermions on $X$ and may admit an interpretation in terms of some random matrix models. The empirical measures, associated with such $\beta$-ensembles, converge almost surely to an equilibrium measure when $p$ goes to infinity. We establish a large deviation theorem (LDT) with an effective speed of convergence for these empirical measures. Our study covers a large class of $\beta$-ensembles on a compact subset of the unit sphere $\mathbb {S}^n\subset \mathbb {R}^{n+1}$ or of the Euclidean space $\mathbb {R}^n$.
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Additional Information
  • Tien-Cuong Dinh
  • Affiliation: Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076
  • MR Author ID: 608547
  • Email:
  • Viêt-Anh Nguyên
  • Affiliation: Université de Lille 1, Laboratoire de mathématiques Paul Painlevé, CNRS U.M.R. 8524, 59655 Villeneuve d’Ascq Cedex, France
  • Email:
  • Received by editor(s): April 10, 2016
  • Received by editor(s) in revised form: December 20, 2016
  • Published electronically: February 26, 2018
  • Additional Notes: The first author was supported by Start-Up Grant R-146-000-204-133 from National University of Singapore
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 6565-6584
  • MSC (2010): Primary 32U15; Secondary 32L05, 60F10
  • DOI:
  • MathSciNet review: 3814341