论文标题
在均匀为突系统的几何形状上
On the geometry of uniform meandric systems
论文作者
论文摘要
一个尺寸$ n $的曲折系统是$ \ {1，\ dots，2n \} $的两个弧形图（非交叉完美匹配）形成的循环集，一个是在实际线上绘制的，另一个绘制在真实行之上。均匀的随机为突系统可以看作是由哈密顿路径（对应于真实线）和循环（由弧形形成）装饰的随机平面图。根据物理启发式和数值证据，我们猜测，该装饰的随机平面图的缩放限制由由liouville量子重力（LQG）带有参数$γ= \ sqrt 2 $的独立三倍（LQG），schramm-loewner-loewner-loewner Evolution（sle）与参数$κ= 8 $ cle $ commartomal commartomal uncomement comminter = 8 $ comparome commartal = 我们证明了几个与此猜想一致的严格结果。特别是，一个均匀的为地系统接收几乎宏观的图形距离直径的循环，概率很高。此外，A.S.，具有边界的统一无限为突系统没有无限的路径。但是，A.S.，其边界修饰的版本具有独特的无限路径，其缩放限制被认为是弦SLE $ _6 $。
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$_6$.
