arXiv 1111.72272011-11math.PR

带边界随机平面四边形剖分的标度极限

Scaling Limit of Random Planar Quadrangulations with a Boundary

Jérémie Bettinelli

摘要：

本文讨论了边界长度为面数平方根数量级的平面四边形剖分的尺度极限。我们考虑整数序列$(σ_n)$使得$σ_n/\sqrt{2n}$趋向于某个$σ\in[0,\infty]$ .对于每个$n \ge 1$，我们称$q_n$为均匀分布在所有有根平面四边形的集合上的随机映射，其边界具有$n$面和边界上的$2σ_n$半边。对于$σ\in (0,\infty)$，我们将$q_n$视为度量空间，通过赋予其顶点集以由$n^{-1/4}$重新缩放的图度量。我们表明，这个度量空间收敛分布，至少沿着一些子序列，向一个有限的随机度量空间，在这个意义上的Gromov-Hausdorff拓扑。我们证明了极限度量空间几乎必然是一个Hausdorff维数为4的空间，其边界Hausdorff维数为2，与二维圆盘同胚。对于$σ=0$，相同的收敛性在没有提取的情况下也成立，极限是所谓的布朗映射。对于$σ=\infty$，适当的缩放成为$σ_n^{-1/2}$，我们得到了一个收敛到奥尔德斯的CRT。

英文摘要：

We discuss the scaling limit of large planar quadrangulations with a boundary whose length is of order the square root of the number of faces. We consider a sequence $(σ_n)$ of integers such that $σ_n/\sqrt{2n}$ tends to some $σ\in[0,\infty]$. For every $n \ge 1$, we call $q_n$ a random map uniformly distributed over the set of all rooted planar quadrangulations with a boundary having $n$ faces and $2σ_n$ half-edges on the boundary. For $σ\in (0,\infty)$, we view $q_n$ as a metric space by endowing its set of vertices with the graph metric, rescaled by $n^{-1/4}$. We show that this metric space converges in distribution, at least along some subsequence, toward a limiting random metric space, in the sense of the Gromov--Hausdorff topology. We show that the limiting metric space is almost surely a space of Hausdorff dimension 4 with a boundary of Hausdorff dimension 2 that is homeomorphic to the two-dimensional disc. For $σ=0$, the same convergence holds without extraction and the limit is the so-called Brownian map. For $σ=\infty$, the proper scaling becomes $σ_n^{-1/2}$ and we obtain a convergence toward Aldous's CRT.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文