Gaussian beta ensembles : The perfect freezing transition and its characterization in terms of Beurling–Landau densities
(2026) In Annales de l'institut Henri Poincare (B) Probability and Statistics 62(1). p.296-327- Abstract
The Gaussian β-ensemble is a real n-point configuration {xj}n1 picked randomly with respect to the Boltzmann factor e− β 2 Hn, where Hn = ∑i≠j log |xi−1xj| + n ∑ni=112 xi2. It is well known that the point process {xj }n1 tends to follow the semicircle law σ (x) = 21π√(4 − x2 )+ in certain average senses. A Fekete configuration (minimizer of Hn) is spread out in a much more uniform way in the interval [−2, 2] with respect to... (More)
The Gaussian β-ensemble is a real n-point configuration {xj}n1 picked randomly with respect to the Boltzmann factor e− β 2 Hn, where Hn = ∑i≠j log |xi−1xj| + n ∑ni=112 xi2. It is well known that the point process {xj }n1 tends to follow the semicircle law σ (x) = 21π√(4 − x2 )+ in certain average senses. A Fekete configuration (minimizer of Hn) is spread out in a much more uniform way in the interval [−2, 2] with respect to the 1 regularization σn(x) = max{σ (x), n− 3 } of the semicircle law. In particular, Fekete configurations are “equidistributed” with respect to σn(x), in a certain technical sense of Beurling–Landau densities. We consider the problem of characterizing sequences βn of inverse temperatures, which guarantee almost sure equidistribution as n → ∞. We find that a necessary and sufficient condition is that βn grows at least logarithmically in n: βn ≳ log n. We call this growth rate the perfect freezing regime. Along the way, we give several further results on the distribution of particles when βn ≳ log n, for example on minimal spacing and discrepancies, and that with high probability a random sample solves certain sampling and interpolation problems for weighted polynomials. (In this context, Fekete sets correspond to β ≡ ∞.) The condition βn ≳ log n was introduced in earlier works due to some of the authors in the context of two-dimensional Coulomb gas ensembles, where it is shown to be sufficient for equidistribution. Interestingly, although the technical implementation requires some considerable modifications, the strategy from dimension two adapts well to prove sufficiency also for one-dimensional Gaussian ensembles. On a technical level, we use estimates for weighted polynomials due to Levin, Lubinsky, Gustavsson and others. The other direction (necessity) involves estimates due to Ledoux and Rider on the distribution of particles which fall near or outside the boundary.
(Less)
- author
- Ameur, Yacin LU ; Marceca, Felipe and Romero, José Luis
- organization
- publishing date
- 2026-02
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Discrepancy, Equidistribution, Gaussian ensemble, Perfect freezing, Random quadrature, Separation
- in
- Annales de l'institut Henri Poincare (B) Probability and Statistics
- volume
- 62
- issue
- 1
- pages
- 32 pages
- publisher
- Institute of Mathematical Statistics
- external identifiers
-
- scopus:105032424612
- ISSN
- 0246-0203
- DOI
- 10.1214/24-AIHP1524
- language
- English
- LU publication?
- yes
- id
- e224162c-fa69-4227-8c11-1407e282aa28
- date added to LUP
- 2026-04-28 16:03:23
- date last changed
- 2026-04-28 16:03:51
@article{e224162c-fa69-4227-8c11-1407e282aa28,
abstract = {{<p>The Gaussian β-ensemble is a real n-point configuration {x<sub>j</sub>}<sup>n</sup><sub>1</sub> picked randomly with respect to the Boltzmann factor e<sup>− β 2 Hn</sup>, where H<sub>n</sub> = <sup>∑</sup><sub>i</sub>≠<sub>j</sub> log <sub>|xi−</sub><sup>1</sup><sub>xj</sub><sub>|</sub> + n <sup>∑n</sup><sub>i</sub><sub>=1</sub><sup>1</sup><sub>2</sub> x<sub>i</sub><sup>2</sup>. It is well known that the point process {x<sub>j</sub> }<sup>n</sup><sub>1</sub> tends to follow the semicircle law σ (x) = <sub>2</sub><sup>1</sup><sub>π</sub><sup>√</sup>(4 − x<sup>2</sup> )<sub>+</sub> in certain average senses. A Fekete configuration (minimizer of H<sub>n</sub>) is spread out in a much more uniform way in the interval [−2, 2] with respect to the 1 regularization σ<sub>n</sub>(x) = max{σ (x), n<sup>− 3</sup> } of the semicircle law. In particular, Fekete configurations are “equidistributed” with respect to σ<sub>n</sub>(x), in a certain technical sense of Beurling–Landau densities. We consider the problem of characterizing sequences β<sub>n</sub> of inverse temperatures, which guarantee almost sure equidistribution as n → ∞. We find that a necessary and sufficient condition is that β<sub>n</sub> grows at least logarithmically in n: β<sub>n</sub> ≳ log n. We call this growth rate the perfect freezing regime. Along the way, we give several further results on the distribution of particles when β<sub>n</sub> ≳ log n, for example on minimal spacing and discrepancies, and that with high probability a random sample solves certain sampling and interpolation problems for weighted polynomials. (In this context, Fekete sets correspond to β ≡ ∞.) The condition β<sub>n</sub> ≳ log n was introduced in earlier works due to some of the authors in the context of two-dimensional Coulomb gas ensembles, where it is shown to be sufficient for equidistribution. Interestingly, although the technical implementation requires some considerable modifications, the strategy from dimension two adapts well to prove sufficiency also for one-dimensional Gaussian ensembles. On a technical level, we use estimates for weighted polynomials due to Levin, Lubinsky, Gustavsson and others. The other direction (necessity) involves estimates due to Ledoux and Rider on the distribution of particles which fall near or outside the boundary.</p>}},
author = {{Ameur, Yacin and Marceca, Felipe and Romero, José Luis}},
issn = {{0246-0203}},
keywords = {{Discrepancy; Equidistribution; Gaussian ensemble; Perfect freezing; Random quadrature; Separation}},
language = {{eng}},
number = {{1}},
pages = {{296--327}},
publisher = {{Institute of Mathematical Statistics}},
series = {{Annales de l'institut Henri Poincare (B) Probability and Statistics}},
title = {{Gaussian beta ensembles : The perfect freezing transition and its characterization in terms of Beurling–Landau densities}},
url = {{http://dx.doi.org/10.1214/24-AIHP1524}},
doi = {{10.1214/24-AIHP1524}},
volume = {{62}},
year = {{2026}},
}