中间温度下 $\beta$-系综的介观相变
Mesoscopic transition for $β$-ensembles at intermediary temperature
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- KTH Royal Institute of Technology（瑞典皇家理工学院）
- College of the Holy Cross（圣十字学院）
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中文总结 AI 辅助
本文针对 $\beta$-系综在中间温度区间建立介观中心极限定理，揭示随机矩阵与泊松区间之间的临界相变，并给出自由能展开。
中文摘要 AI 辅助
本文建立了 $\beta$-系综（或对数气体）线性统计量的介观中心极限定理，当维数 $N\to\infty$ 时，在温度区间 $1/N\ll\beta(N)\le 1$ 内成立。为简化起见，我们假设势函数是单割正则且解析的。在该区间内，波动的大小依赖于 $\beta(N)$ 和介观尺度。我们证明在临界值 $\eta\asymp 1/ N\beta(N)$ 处存在一个相变，介于随机矩阵区间（其极限方差由 $\mathsf{H}^{1/2}$-范数给出）和泊松区间（其极限方差由 $L^2$-范数给出）之间。我们还描述了临界区间。该中心极限定理的证明依赖于中间温度下的最优局部律和 $\beta$-系综的 Stein 方法。特别地，在该区间内，有必要构造经典平衡测度的新修正项，以获得线性统计量的适当重新中心化并描述其波动。我们还得到了自由能展开式。
英文摘要
This paper establishes a mesoscopic central limit theorem for linear statistics of $β$-ensembles or log-gas, as the dimension $N\to\infty$, in the temperature regime $1/N\llβ(N)\le 1$. 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 $η\asymp 1/ Nβ(N)$ between a Random Matrix regime, where the limiting variance is given by the $\mathsf{H}^{1/2}$-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.
