The mesoscopic geometry of sparse random maps

N. Curien, I. Kortchemski, Cyril Marzouk
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引用次数: 4

Abstract

We investigate the structure of large uniform random maps with $n$ edges, $\mathrm{f}_n$ faces, and with genus $\mathrm{g}_n$ in the so-called sparse case, where the ratio between the number vertices and edges tends to $1$. We focus on two regimes: the planar case $(\mathrm{f}_n, 2\mathrm{g}_n) = (\mathrm{s}_n, 0)$ and the unicellular case with moderate genus $(\mathrm{f}_n, 2 \mathrm{g}_n) = (1, \mathrm{s}_n-1)$, both when $1 \ll \mathrm{s}_n \ll n$. Albeit different at first sight, these two models can be treated in a unified way using a probabilistic version of the classical core-kernel decomposition. In particular, we show that the number of edges of the core of such maps, obtained by iteratively removing degree $1$ vertices, is concentrated around $\sqrt{n \mathrm{s}_{n}}$. Further, their kernel, obtained by contracting the vertices of the core with degree $2$, is such that the sum of the degree of its vertices exceeds that of a trivalent map by a term of order $\sqrt{\mathrm{s}_{n}^{3}/n}$; in particular they are trivalent with high probability when $\mathrm{s}_{n} \ll n^{1/3}$. This enables us to identify a mesoscopic scale $\sqrt{n/\mathrm{s}_n}$ at which the scaling limits of these random maps can be seen as the local limit of their kernels, which is the dual of the UIPT in the planar case and the infinite three-regular tree in the unicellular case, where each edge is replaced by an independent (biased) Brownian tree with two marked points.
稀疏随机映射的介观几何
在所谓的稀疏情况下,我们研究了具有$n$边,$\mathrm{f}_n$面和$\mathrm{g}_n$的大型均匀随机映射的结构,其中顶点数与边数之间的比率趋于$1$。我们关注两种情况:平面情况$(\mathrm{f}_n, 2\mathrm{g}_n) = (\mathrm{s}_n, 0)$和中等属情况$(\mathrm{f}_n, 2\mathrm{g}_n) = (1, \mathrm{s}_n-1)$,两者都是$1 \ll \mathrm{s}_n \ll n$。虽然乍一看不同,但这两个模型可以用经典核-核分解的概率版本统一处理。特别地,我们证明了这种映射的核心边的数量,通过迭代地去除次数$1$的顶点,集中在$\sqrt{n \mathrm{s}_{n}}$附近。进一步,它们的核,通过核的顶点与度$2$的缩并得到,使得其顶点的度数之和超过三价映射的度数之和为$\sqrt{\mathrm{s}_{n}^{3}/n}$阶项;特别是当$\ mathm {s}_{n} \ll n^{1/3}$时,它们是高概率的三价。这使我们能够识别一个介观尺度$\sqrt{n/\ maththrm {s}_n}$,在这个尺度上,这些随机映射的缩放极限可以看作是它们核的局部极限,这是平面情况下的upt和单细胞情况下的无限三规则树的对偶,其中每个边都被一个独立的(有偏的)布朗树取代,其中有两个标记点。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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