超临界Liouville第一通道渗流的紧密性
Tightness of supercritical Liouville first passage percolation
摘要：
具有参数$ξ>0$的Liouville首次通过渗流（LFPP）是通过沿路径积分$e^{ξh_ε}$而得到的平面上的随机距离函数族$\{D_h^ε\}_{ε>0}$，其中$ε>0$的$h_ε$是平面高斯自由场的光滑软化。Ding-Dubédat-Dunlap-Falconet和Gwynne-Miller以前的工作表明，存在一个临界值$ξ_{\mathrm{crit}} > 0$，使得对于$ξ< ξ_{\mathrm{crit}}$，LFPP在适当的重新缩放下收敛到平面上的随机度量，该随机度量引起与欧几里得度量相同的拓扑（对于$γ= γ(ξ)\in (0,2)$，所谓的$γ$ - \emph{Liouville quantum gravity metric}）。我们表明，对于所有的$ξ> 0$，LFPP度量是紧的拓扑结构上较低的连续函数。对于$ξ> ξ_{\mathrm{crit}}$，每一个可能的连续极限$D_h$都是平面上的度量，这使得\emph{not}诱导欧几里得拓扑：相反，存在不可数的、稠密的、勒贝格测度-零点集$z\in\mathbb C $，使得对于每一个$w\in\mathbb C\setminus \{z\}$，$D_h(z,w) = \infty$。我们期望这些基本的极限度规与$(1,25)$中的具有物质中心电荷的Liouville量子引力有关。
英文摘要：
Liouville first passage percolation (LFPP) with parameter $ξ>0$ is the family of random distance functions $\{D_h^ε\}_{ε>0}$ on the plane obtained by integrating $e^{ξh_ε}$ along paths, where $h_ε$ for $ε>0$ is a smooth mollification of the planar Gaussian free field. Previous work by Ding-Dubédat-Dunlap-Falconet and Gwynne-Miller has shown that there is a critical value $ξ_{\mathrm{crit}} > 0$ such that for $ξ< ξ_{\mathrm{crit}}$, LFPP converges under appropriate re-scaling to a random metric on the plane which induces the same topology as the Euclidean metric (the so-called $γ$-\emph{Liouville quantum gravity metric} for $γ= γ(ξ)\in (0,2)$). We show that for all $ξ> 0$, the LFPP metrics are tight with respect to the topology on lower semicontinuous functions. For $ξ> ξ_{\mathrm{crit}}$, every possible subsequential limit $D_h$ is a metric on the plane which does \emph{not} induce the Euclidean topology: rather, there is an uncountable, dense, Lebesgue measure-zero set of points $z\in\mathbb C $ such that $D_h(z,w) = \infty$ for every $w\in\mathbb C\setminus \{z\}$. We expect that these subsequential limiting metrics are related to Liouville quantum gravity with matter central charge in $(1,25)$.
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