Mesoscopic transition for β-ensembles at intermediary temperature
Abstract
This paper establishes a mesoscopic central limit theorem for linear statistics of β-ensembles or log-gas, as the dimension N→∞, in the temperature regime 1/N≪β(N)≤ 1. For simplicity, we assume that the potential is one-cut regular and analytic. In this regime, the size of the fluctuations depends on β(N) and the mesoscopic scale. We show that there is a transition at a critical ηasymp 1/ Nβ(N) between a Random Matrix regime, where the limiting variance is given by the H1/2-norm and a Poisson regime where the limiting variance is given by the L2-norm. We also describe the critical regime. The proof of the CLT relies on optimal local laws at intermediate temperatures and Stein's method for β-ensembles. In particular, in this regime, it is necessary to construct new correction terms to the classical equilibrium measure to obtain a suitable re-centring of linear statistics and describe their fluctuations. We also obtain a free energy expansion.
Related
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BibTeX
@misc{guera2026mesoscopic,
title = {{Mesoscopic transition for $β$-ensembles at intermediary temperature}},
author = {Charlie Dworaczek Guera and Gaultier Lambert and Luke Peilen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25301},
eprint = {2609.25301},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25301},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Mesoscopic transition for β-ensembles at intermediary temperature
AU - Charlie Dworaczek Guera
AU - Gaultier Lambert
AU - Luke Peilen
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.25301
UR - https://arxiv.org/abs/2609.25301
AB - This paper establishes a mesoscopic central limit theorem for linear statistics of β-ensembles or log-gas, as the dimension N→∞, in the temperature regime 1/N≪β(N)≤ 1. For simplicity, we assume that the potential is one-cut regular and analytic. In this regime, the size of the fluctuations depends on β(N) and the mesoscopic scale. We show that there is a transition at a critical ηasymp 1/ Nβ(N) between a Random Matrix regime, where the limiting variance is given by the H^1/2-norm and a Poisson regime where the limiting variance is given by the L^2-norm. We also describe the critical regime. The proof of the CLT relies on optimal local laws at intermediate temperatures and Stein's method for β-ensembles. In particular, in this regime, it is necessary to construct new correction terms to the classical equilibrium measure to obtain a suitable re-centring of linear statistics and describe their fluctuations. We also obtain a free energy expansion.
ER -