论文标题
二维离散高斯自由场的接近最大马克西马
Near-maxima of the two-dimensional Discrete Gaussian Free Field
论文作者
论文摘要
我们考虑域中的离散高斯免费字段（DGFF）$ d_n \ subseteq \ subseteq \ mathbb z^2 $从nice域$ d \ subseteq \ subseteq \ mathbb r^2 $产生的缩放量。我们研究了以下绝对最大值的值$ \ sqrt {\ log n} $的统计信息。 Encoded as a point process on $D\times\mathbb R$, the scaled spatial distribution of these near-extremal level sets in $D_N$ and the field values (in units of $\sqrt{\log N}$ below the absolute maximum) tends, as $N\to\infty$, in law to the product of the critical Liouville Quantum Gravity (cLQG) $Z^D$ and the瑞利法律。融合与极端过程共同存在，为此，$ z^d $作为限制泊松点过程的强度度量以及DGFF本身的强度度量。然后由限制字段定义的CLQG与$ z^d $重合。虽然相对于极限连续GFF的极限近极过程是可以测量的，但极限的极限过程却不是。我们的结果解释了为什么“标准”晶格CLQG测量的各种方法导致相同的极限对象，Modulo总体归一化。
We consider the Discrete Gaussian Free Field (DGFF) in domains $D_N\subseteq\mathbb Z^2$ arising, via scaling by $N$, from nice domains $D\subseteq\mathbb R^2$. We study the statistics of the values order $\sqrt{\log N}$ below the absolute maximum. Encoded as a point process on $D\times\mathbb R$, the scaled spatial distribution of these near-extremal level sets in $D_N$ and the field values (in units of $\sqrt{\log N}$ below the absolute maximum) tends, as $N\to\infty$, in law to the product of the critical Liouville Quantum Gravity (cLQG) $Z^D$ and the Rayleigh law. The convergence holds jointly with the extremal process, for which $Z^D$ enters as the intensity measure of the limiting Poisson point process, and that of the DGFF itself; the cLQG defined by the limit field then coincides with $Z^D$. While the limit near-extremal process is measurable with respect to the limit continuum GFF, the limit extremal process is not. Our results explain why the various ways to "norm" the lattice cLQG measure lead to the same limit object, modulo overall normalization.
