论文标题
高斯beta合奏：完美的冻结过渡及其在beurling-landau密度方面的特征
Gaussian beta ensembles: the perfect freezing transition and its characterization in terms of Beurling-Landau densities
论文作者
论文摘要
高斯$β$ - 填充是一个真正的$ n $ - 点配置$ \ {x_j \} _ 1^n $相对于Boltzmann因子$ e^{ - \fracβ2H_n} $，随机挑选的，$ e^{ - \fracβ2h_n} $，$ h_n = \ sum_ = \ sum_ {i \ ne j} 1 {| x_i-x_j |}+n \ sum_ {i = 1}^n \ tfrac 12x_i^2. $点过程$ \ {x_j \} _ 1^n $倾向于遵循半圆法$ $之一$ = $σ（x）= \ tfrac 1 {x）= \ tfrac 1 {2π} \ s Qrt sqrt^$ sqrt} $ semeres $ semeres $ semeres。 相对于正则化$σ_n（x）= \ max \ {σ（x），n^{ - \ frac 1 3} $，以一种间隔$ [ - 2,2] $，以更均匀的方式以更均匀的方式分散了fekete配置（$ h_n $的最小化）。特别是，从某种技术意义上讲，fekete配置相对于$σ_n（x）$“等级”。 我们考虑表征序列$β_n$的反向温度的问题，这几乎可以确保等均分布为$ n \ to \ infty $。我们发现，必要且充分的条件是$β_n$至少在$ n $：$ n $：$β_n\ gtrsim \ logn。$n。$n。$n。$n。$ n。我们称之为完美的冻结方案。当$β_N\ gtrsim \ log n $（例如，在最小间距，差异以及加权多项式的采样和插值时），我们给出了颗粒分布的进一步结果。 在二维库仑气体集合的背景下，一些作者引入了条件$β_N\ gtrsim \ log n $，在这种情况下，这足以足以进行等均衡。尽管技术实施需要进行一些相当大的修改，但尺寸二的策略很好地适应了一维高斯合奏的充分性。从技术层面上讲，由于Levin，Lubinsky，Gustavsson等，我们对加权多项式使用估算值。另一个方向（必要性）涉及由于Ledoux和Rider引起的估计，这些粒子的分布落在边界附近或外部。
The Gaussian $β$-ensemble is a real $n$-point configuration $\{x_j\}_1^n$ picked randomly with respect to the Boltzmann factor $e^{-\fracβ2H_n}$, $H_n=\sum_{i\ne j}\log\frac 1{|x_i-x_j|}+n\sum_{i=1}^n\tfrac 12x_i^2.$ The point process $\{x_j\}_1^n$ tends to follow the semicircle law $σ(x)=\tfrac 1{2π}\sqrt{(4-x^2)_+}$ in certain average senses. A Fekete configuration (minimizer of $H_n$) is spread out in a much more uniform way in the interval $[-2,2]$ with respect to the regularization $σ_n(x)=\max\{σ(x),n^{-\frac 1 3}\}$ of the semicircle law. In particular, Fekete configurations are "equidistributed" with respect to $σ_n(x)$, in a certain technical sense of Beurling-Landau densities. We consider the problem of characterizing sequences $β_n$ of inverse temperatures, which guarantee almost sure equidistribution as $n\to\infty$. We find that a necessary and sufficient condition is that $β_n$ grows at least logarithmically in $n$: $$β_n\gtrsim \log n.$$ We call this growth rate the perfect freezing regime. We give several further results on the distribution of particles when $β_n\gtrsim\log n$, for example on minimal spacing, discrepancies, and sampling and interpolation for weighted polynomials. The condition $β_n\gtrsim\log n$ was introduced by some of the authors in the context of two-dimensional Coulomb gas ensembles, where it is shown to be sufficient for equidistribution. 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.
