关于均匀蜿蜒系统几何性质的研究
On the geometry of uniform meandric systems
AI总结：
本文研究均匀蜿蜒系统的几何性质，提出其标度极限由LQG球面、SLE曲线和CLE环系综组成，并证明相关严格结果，包括近宏观直径环的存在性和边界修正版本的唯一无限路径。
摘要：
大小为 $n$ 的蜿蜒系统是由定义在集合 $\{1,\dots,2n\}$ 上的两个弧图（非交叉完美匹配）所形成的环的集合，其中一个弧图绘制在实线上方，另一个绘制在实线下方。均匀随机蜿蜒系统可以被视为一个由哈密顿路径（对应于实线）和一组环（由弧形成）装饰的随机平面地图。基于物理学启发式方法和数值证据，我们推测这种装饰随机平面地图的标度极限由一个独立三元组给出，该三元组由参数 $\gamma=\sqrt 2$ 的刘维尔量子引力（LQG）球面、参数 $\kappa=8$ 的施拉姆-勒夫纳演化（SLE）曲线以及参数 $\kappa=6$ 的共形环系综（CLE）组成。我们证明了几个与该推测一致的严格结果。特别地，一个均匀蜿蜒系统以高概率包含具有近宏观图距离直径的环。此外，几乎必然地，具有边界的均匀无限蜿蜒系统没有无限路径。但是，几乎必然地，其边界修正版本具有唯一的无限路径，其标度极限被推测为弦 SLE$_6$。
英文摘要：
A meandric system of size $n$ is the set of loops formed from two arc diagrams (non-crossing perfect matchings) on $\{1,\dots,2n\}$, one drawn above the real line and the other below the real line. A uniform random meandric system can be viewed as a random planar map decorated by a Hamiltonian path (corresponding to the real line) and a collection of loops (formed by the arcs). Based on physics heuristics and numerical evidence, we conjecture that the scaling limit of this decorated random planar map is given by an independent triple consisting of a Liouville quantum gravity (LQG) sphere with parameter $γ=\sqrt 2$, a Schramm-Loewner evolution (SLE) curve with parameter $κ=8$, and a conformal loop ensemble (CLE) with parameter $κ=6$. We prove several rigorous results which are consistent with this conjecture. In particular, a uniform meandric system admits loops of nearly macroscopic graph-distance diameter with high probability. Furthermore, a.s., the uniform infinite meandric system with boundary has no infinite path. But, a.s., its boundary-modified version has a unique infinite path whose scaling limit is conjectured to be chordal SLE$_6$.
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