二维Ginzburg-Landau场极大值的次序紧性

Subsequential tightness for branching random walk in random environment

We consider branching random walk in random environment (BRWRE) and prove the existence of deterministic subsequences along which their maximum, centered at its mean, is tight. This partially answers an open question in arXiv:1711.00852. The method of proof adapts an argument developed by Dekking and Host for branching random walks with bounded increments. The question of tightness without the need for subsequences remains open.

二维均匀生成树上简单随机游动的序贯标度极限

Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree

本文的第一个主要结果是：在元素被测度的空间中，（重标度）二维一致生成树的律是紧的，有根实树连续嵌入欧几里得空间。得到了该空间的内在度量、测度和极限嵌入的各种性质，特别证明了任一极限在其内在度量中的豪斯多夫维数几乎必然等于$8/5$ .此外，应用紧性结果，证明了二维均匀生成树上的简单随机游动的退火律在适当的重标度下是紧的。对于嵌入到欧几里得空间中的随机实树上的扩散过程，得到了详细的转移密度估计。

Ginzburg-Landau 场最大值的紧性

Tightness of the maximum of Ginzburg-Landau fields

本文在临界维数 $d=2$ 下，对满足一致凸性条件的离散 Ginzburg-Landau 场，证明了其边长为 $N$ 的盒子上的最大值经显式中心化后具有紧性。

Tightness of approximations to the chemical distance metric for simple conformal loop ensembles

Suppose that $Γ$ is a conformal loop ensemble (CLE$_κ$) with simple loops ($κ\in (8/3,4)$) in a simply connected domain $D \subseteq {\mathbf C}$ whose boundary is itself a type of CLE$_κ$ loop. Let $Υ$ be the carpet of $Γ$, i.e., the set of points in $D$ not surrounded by a loop of $Γ$. We prove that certain approximations to the chemical distance metric in $Υ$ are tight. More precisely, for each path $ω\colon [0,1] \to Υ$ and $ε> 0$ we let ${\mathfrak N}_ε(ω)$ be the Lebesgue measure of the $ε$-neighborhood of $ω$. For $z,w \in Υ$ we let ${\mathfrak d}_ε(z,w;Γ) = \inf_ω{\mathfrak N}_ε(ω)$ wh…

任意维度下对数相关高斯场指数度量的紧致性

Tightness of exponential metrics for log-correlated Gaussian fields in arbitrary dimension

研究\(\mathbb{R}^d\)（\(d\geq2\)）上类似刘维尔量子引力度量的自然近似方案，通过特定高斯随机函数序列构建随机度量族，证明在亚临界阶段经重标度后度量紧致，子序列极限诱导欧几里得拓扑，并列出诸多开放问题。

微观减速高斯场的极值

Extrema of cooling branching Brownian motion and related Gaussian fields

引入一族协方差结构具有非均匀微观减速的高斯场，研究其最大值的渐近行为，发现一阶增长指数随参数α线性变化，而二阶修正项在α=1/3处发生相变。