论文标题
liouville首次通过渗透的距离指数为正
The distance exponent for Liouville first passage percolation is positive
论文作者
论文摘要
带有参数$ξ> 0 $的离散Liouville First Passage Percolation（LFPP）是$ \ Mathbb z^2 $的子图上的随机度量，通过分配每个顶点$ z $ a $ e^e^{ξh（z）} $，其中$ h $是$ e^{eCy^{eC^{eCy^{e^{n $ h $是离散的高斯免费场。我们表明，离散LFPP的距离指数严格呈$ξ> 0 $。更确切地说，对于指数$α> 0 $，取决于$ξ$，对于$ 2^n $的离散环的内边界和外部边界之间的离散LFPP距离通常至少为$ 2^{αn} $。这是一个至关重要的输入，证明了LFPP接受所有$ξ> 0 $的非平凡的后续缩放限制，并且对Liouville量子重力的距离研究也具有理论意义。
Discrete Liouville first passage percolation (LFPP) with parameter $ξ> 0$ is the random metric on a sub-graph of $\mathbb Z^2$ obtained by assigning each vertex $z$ a weight of $e^{ξh(z)}$, where $h$ is the discrete Gaussian free field. We show that the distance exponent for discrete LFPP is strictly positive for all $ξ> 0$. More precisely, the discrete LFPP distance between the inner and outer boundaries of a discrete annulus of size $2^n$ is typically at least $2^{αn}$ for an exponent $α> 0$ depending on $ξ$. This is a crucial input in the proof that LFPP admits non-trivial subsequential scaling limits for all $ξ> 0$ and also has theoretical implications for the study of distances in Liouville quantum gravity.
