Limit theorems for biorthogonal ensembles and related combinatorial identities

Lambert, Gaultier

doi:10.1016/j.aim.2017.12.025

Cited by 30 publications

(28 citation statements)

References 43 publications

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“…In the special case β = 2, there are also other types of proofs which rely on the determinantal structure of the ensembles P N V,2 and are valid in greater generality, see e.g. [8] or [18].…”

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confidence: 99%

Quantitative normal approximation of linear statistics of $\beta$-ensembles

Lambert¹,

Ledoux²,

Webb³

2019

Ann. Probab.

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We present a new approach, inspired by Stein's method, to prove a central limit theorem (CLT) for linear statistics of βensembles in the one-cut regime. Compared with the previous proofs, our result requires less regularity on the potential and provides a rate of convergence in the quadratic Kantorovich or Wasserstein-2 distance. The rate depends both on the regularity of the potential and the test functions, and we prove that it is optimal in the case of the Gaussian Unitary Ensemble (GUE) for certain polynomial test functions.The method relies on a general normal approximation result of independent interest which is valid for a large class of Gibbs-type distributions. In the context of β-ensembles, this leads to a multidimensional CLT for a sequence of linear statistics which are approximate eigenfunctions of the infinitesimal generator of Dyson Brownian motion once the various error terms are controlled using the rigidity results of Bourgade, Erdős and Yau.

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confidence: 99%

Quantitative normal approximation of linear statistics of $\beta$-ensembles

Lambert¹,

Ledoux²,

Webb³

2019

Ann. Probab.

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“…In particular they obtain central limit theorems for the fluctuations of the linear statistics when convergence of the recurrence coefficients is assumed. These results have been recovered by Lambert [2015] using technics more in the spirit of this note, namely involving sums over paths weighted by the recurrence coefficients.…”

Section: Higher Order Cumulants and Fluctuationsmentioning

confidence: 95%

“…In Section 5, we briefly present the recent results of Breuer and Duits [2017] and Lambert [2015] on fluctuations for the linear statistics of polynomial ensembles satisfying a finite-term recurrence relation.…”

Section: Organisationmentioning

confidence: 99%

Polynomial Ensembles and Recurrence Coefficients

Hardy

2017

Constr Approx

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Polynomial ensembles are determinantal point processes associated with (non necessarily orthogonal) projections onto polynomial subspaces. The aim of this survey article is to put forward the use of recurrence coefficients to obtain the global asymptotic behavior of such ensembles in a rather simple way. We provide a unified approach to recover well-known convergence results for real OP ensembles. We study the mutual convergence of the polynomial ensemble and the zeros of its average characteristic polynomial; we discuss in particular the complex setting. We also control the variance of linear statistics of polynomial ensembles and derive comparison results, as well as asymptotic formulas for real OP ensembles. Finally, we reinterpret the classical algorithm to sample determinantal point processes so as to cover the setting of non-orthogonal projection kernels. A few open problems are also suggested.

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“…The convergence D(ξ N g i ) → D(ξ g i ) was proven by Spohn [42] and Soshnikov [40,41]. For further developments see also works [21,25,26,7,8], where certain Central Limit Theorems were established for linear statistics with bounded variance, related to the marginal convergence D(ξ N g i ) → D(ξ g i ). More precisely, in [21,25] and [7] the Central Limit Theorems were proven for linear statistics of various orthogonal polynomial ensembles on mesoscopic scales.…”

Section: Central Limit Theorem For Linear Statisticsmentioning

confidence: 99%

“…More precisely, in [21,25] and [7] the Central Limit Theorems were proven for linear statistics of various orthogonal polynomial ensembles on mesoscopic scales. In [26] and [8] those were obtained for linear statistics of certain biorthogonal ensembles.…”

Section: Central Limit Theorem For Linear Statisticsmentioning

confidence: 99%

A Functional Limit Theorem for the Sine-Process

Bufetov

Dymov

2018

International Mathematics Research Notices

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The main result of this paper is a functional limit theorem for the sine-process. In particular, we study the limit distribution, in the space of trajectories, for the number of particles in a growing interval. The sine-process has the Kolmogorov property and satisfies the Central Limit Theorem, but our functional limit theorem is very different from the Donsker Invariance Principle. We show that the time integral of our process can be approximated by the sum of a linear Gaussian process and independent Gaussian fluctuations whose covariance matrix is computed explicitly. We interpret these results in terms of the Gaussian Free Field convergence for the random matrix models. The proof relies on a general form of the multidimensional Central Limit Theorem under the sine-process for linear statistics of two types: those having growing variance and those with bounded variance corresponding to observables of Sobolev regularity 1/2. Contents

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Limit theorems for biorthogonal ensembles and related combinatorial identities

Cited by 30 publications

References 43 publications

Quantitative normal approximation of linear statistics of $\beta$-ensembles

Quantitative normal approximation of linear statistics of $\beta$-ensembles

Polynomial Ensembles and Recurrence Coefficients

A Functional Limit Theorem for the Sine-Process

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