论文标题
梯度场模型的局部中央限制定理
Local central limit theorem for gradient field models
论文作者
论文摘要
我们考虑$ \ left [-n，n \ right] ^{2} \ cap \ mathbb {z} ^{2} $中的梯度字段模型，并具有均匀的凸相互潜力。 naddaf-spencer \ cite {ns}和米勒\ cite {mi}证明了该场的线性统计量的宏观平均值会收敛到连续的高斯自由场。在本文中，我们证明了$ ϕ（0）/\ sqrt {\ log n} $的分布均匀收敛到高斯密度，并具有浆果 - 埃塞恩类型的绑定。这意味着$ ϕ（0）$的分布在$ [ - \ sqrt {\ log n}，\ sqrt {\ log n}] $之间足够``高斯''。
We consider the gradient field model in $\left[ -N,N\right] ^{2}\cap \mathbb{Z}^{2}$ with a uniformly convex interaction potential. Naddaf-Spencer \cite{NS} and Miller \cite{Mi} proved that the macroscopic averages of linear statistics of the field converge to a continuum Gaussian free field. In this paper we prove the distribution of $ϕ(0)/\sqrt{\log N}$ converges uniformly to a Gaussian density, with a Berry-Esseen type bound. This implies the distribution of $ϕ(0)$ is sufficiently `Gaussian like' between $[-\sqrt {\log N}, \sqrt {\log N}]$.
