论文标题
紧凑的布朗表面II。定向表面
Compact Brownian surfaces II. Orientable surfaces
论文作者
论文摘要
修复具有边界的任意紧凑定位表面，并考虑具有$ n $ faces和边界组件$ \ sqrt n $的均匀两分随机四边形或较低顺序的边界组件长度。通过$ n^{ - 1/4} $重归于的通常的图形度量，将其标记在每个边界组件上，并在其顶点集合上以$ n^{ - 1} $重归其为基础上的计数量度，并以$ n^{ - 1/1/1/1/1/1/1/1/1/1/1/1/1/1/1的计数测量。我们表明，作为$ n \ to \ infty $，这个随机标记的测得的度量空间在Gromov-Hausdorff-Prokhorov拓扑中收敛于分布，朝着一个随机限制标记的标记测得的度量公式空间，称为Brownian Surface。 这将已知的融合结果均匀的随机平面四边形，最多可以将一个边界分量朝布朗尼球体和布朗磁盘扩展到布朗磁盘，并通过考虑对一般紧凑的定向表面的四个二次法。我们的方法包括将布朗表面切成与布朗球体，布朗磁盘及其非伴动类似物，布朗平面和布朗的半平面的基本碎片，并证明这些基本作品的融合结果，这些片段具有独立的兴趣。
Fix an arbitrary compact orientable surface with a boundary and consider a uniform bipartite random quadrangulation of this surface with $n$ faces and boundary component lengths of order $\sqrt n$ or of lower order. Endow this quadrangulation with the usual graph metric renormalized by $n^{-1/4}$, mark it on each boundary component, and endow it with the counting measure on its vertex set renormalized by $n^{-1}$, as well as the counting measure on each boundary component renormalized by $n^{-1/2}$. We show that, as $n\to\infty$, this random marked measured metric space converges in distribution for the Gromov--Hausdorff--Prokhorov topology, toward a random limiting marked measured metric space called a Brownian surface. This extends known convergence results of uniform random planar quadrangulations with at most one boundary component toward the Brownian sphere and toward the Brownian disk, by considering the case of quadrangulations on general compact orientable surfaces. Our approach consists in cutting a Brownian surface into elementary pieces that are naturally related to the Brownian sphere and the Brownian disk and their noncompact analogs, the Brownian plane and the Brownian half-plane, and to prove convergence results for these elementary pieces, which are of independent interest.
