标题： 二维离散GFF的渗透 II. 双侧水平集的连通性性质 显示英文标题

标题： Percolation of discrete GFF in dimension two II. Connectivity properties of two-sided level sets

摘要： 我们研究二维离散高斯自由场(DGFF)的双侧水平集渗透问题。 对于在边长为$N$的盒子$B_N$中定义的 DGFF$\varphi$，我们证明，在高概率下，存在顶点集$z$中的低交叉，对于任何$\varepsilon>0$，其值为$|\varphi(z)|\le\varepsilon\sqrt{\log N}$，而$\varphi$的平均值和最大值分别为$\sqrt{\log N}$和$\log N$的阶。 我们的方法也强烈表明，在$C\sqrt{\log\log N}$以下存在这样的交叉点，对于足够大的$C$。 作为结果，我们也得到了随机游走的厚点集的连通性性质。 我们依赖于DGFF与临界强度$\alpha=1/2$的随机游走环流(RWLS)之间的同构，并进一步将研究扩展到所有亚临界强度$\alpha\in(0,1/2)$的RWLS的占据场。 For the RWLS in $B_N$, we show that for $\lambda$ large enough, there exist low crossings of $B_N$, remaining below $\lambda$, even though the average occupation time is of order $\log N$. Our results thus uncover a non-trivial phase-transition for this highly-dependent percolation model. For both the DGFF and the occupation field of the RWLS, we further show that such low crossings can be found in the "carpet" of the RWLS - the set of vertices which are not in the interior of any cluster of loops. This work is the second part of a series of two papers. It relies heavily on tools and techniques developed for the RWLS in the first part, especially surgery arguments on loops, which were made possible by a separation result in the RWLS. This allowed us, in that companion paper, to derive several useful properties such as quasi-multiplicativity, and obtain a precise upper bound for the probability that two large connected components of loops "almost touch", which is instrumental here. 显示英文摘要

摘要： We study percolation of two-sided level sets for the discrete Gaussian free field (DGFF) in 2D. For a DGFF $\varphi$ defined in a box $B_N$ with side length $N$, we show that with high probability, there exist low crossings in the set of vertices $z$ with $|\varphi(z)|\le\varepsilon\sqrt{\log N}$ for any $\varepsilon>0$, while the average and the maximum of $\varphi$ are of order $\sqrt{\log N}$ and $\log N$, respectively. Our method also strongly suggests the existence of such crossings below $C\sqrt{\log\log N}$, for $C$ large enough. As a consequence, we also obtain connectivity properties of the set of thick points of a random walk. We rely on an isomorphism between the DGFF and the random walk loop soup (RWLS) with critical intensity $\alpha=1/2$, and further extend our study to the occupation field of the RWLS for all subcritical intensities $\alpha\in(0,1/2)$. For the RWLS in $B_N$, we show that for $\lambda$ large enough, there exist low crossings of $B_N$, remaining below $\lambda$, even though the average occupation time is of order $\log N$. Our results thus uncover a non-trivial phase-transition for this highly-dependent percolation model. For both the DGFF and the occupation field of the RWLS, we further show that such low crossings can be found in the "carpet" of the RWLS - the set of vertices which are not in the interior of any cluster of loops. This work is the second part of a series of two papers. It relies heavily on tools and techniques developed for the RWLS in the first part, especially surgery arguments on loops, which were made possible by a separation result in the RWLS. This allowed us, in that companion paper, to derive several useful properties such as quasi-multiplicativity, and obtain a precise upper bound for the probability that two large connected components of loops "almost touch", which is instrumental here.