On intermediate level sets of two-dimensional discrete Gaussian free field
On intermediate level sets of two-dimensional discrete Gaussian free field
复制标题
二维离散高斯自由场的中间水平集
DOI：
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发表时间：
2016
影响因子：
1.5
通讯作者：
O. Louidor
中科院分区：
文献类型：
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作者：
M. Biskup;O. Louidor
我们考虑适当规则连续域 $Dsubsetmathbb C$ 的放大（方格）版本中的离散高斯自由场 (DGFF)，并描述对于任何 $lambdain(0,1)$ 而言，水平集的缩放极限，包括局部结构，其高度增长为 $lambda$ - 绝对最大值高度的倍数。我们证明，在缩放极限下，从该水平集采样的典型点 $x$ 的缩放空间位置根据 $D$ 中的刘维尔量子引力 (LQG) 度量进行分布，参数等于 $lambda$ 乘以其临界值，$x$ 处的场值具有指数强度度量，并且 $x$ 附近的配置按 $x$ 处的值减少，具有固定 DGFF 减少适当倍数潜在核的定律。特别是，适当归一化的水平集总大小定律收敛于 LQG 度量总质量的定律。这大大强化了达维奥早期的结论。
We consider the discrete Gaussian Free Field (DGFF) in scaled-up (square-lattice) versions of suitably regular continuum domains $Dsubsetmathbb C$ and describe the scaling limit, including local structure, of the level sets at heights growing as a $lambda$-multiple of the height of the absolute maximum, for any $lambdain(0,1)$. We prove that, in the scaling limit, the scaled spatial position of a typical point $x$ sampled from this level set is distributed according to a Liouville Quantum Gravity (LQG) measure in $D$ at parameter equal $lambda$-times its critical value, the field value at $x$ has an exponential intensity measure and the configuration near $x$ reduced by the value at $x$ has the law of a pinned DGFF reduced by a suitable multiple of the potential kernel. In particular, the law of the total size of the level set, properly-normalized, converges that that of the total mass of the LQG measure. This sharpens considerably an earlier conclusion by Daviaud.
影响因子：
0.5
作者：
Berestycki, Nathanael
通讯作者：
Berestycki, Nathanael