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Transactions of the American Mathematical Society

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Large deviation principle for some beta ensembles

Authors: Tien-Cuong Dinh and Viêt-Anh Nguyên
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 32U15; Secondary 32L05, 60F10
Published electronically: February 26, 2018
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Abstract: 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

Viêt-Anh Nguyên
Affiliation: Université de Lille 1, Laboratoire de mathématiques Paul Painlevé, CNRS U.M.R. 8524, 59655 Villeneuve d’Ascq Cedex, France

Keywords: $\beta$-ensemble, large deviations, Fekete points, equilibrium measure, Bergman kernel, Bernstein-Markov property.
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
Article copyright: © Copyright 2018 American Mathematical Society

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