For non-negative integers $(d_n(k))_{k \ge 1}$ such that $\sum_{k \ge 1} d_n(k) = n$, we sample a bipartite planar map with $n$ faces uniformly at random amongst those which have $d_n(k)$ faces of degree $2k$ for every $k \ge 1$ and we study its asymptotic behaviour as $n \to \infty$. We prove that the diameter of such maps grow like $σ_n^{1/2}$, where $σ_n^2 = \sum_{k \ge 1} k (k-1) d_n(k)$ is a global variance term. More precisely, we prove that the vertex-set of these maps equipped with the graph distance divided by $σ_n^{1/2}$ and the uniform probability measure always admits subsequential limits in the Gromov-Hausdorff-Prokhorov topology.
Our proof relies on a bijection with random labelled trees; we are able to prove that the label process is always tight when suitably rescaled, even if the underlying tree is not tight for the Gromov-Hausdorff topology. We also rely on a new spinal decomposition which is of independent interest. Finally this paper also serves as a toolbox for a companion paper in which we discuss more precisely Brownian limits of such maps.
Enhanced Dissipation by Transport Noise for Stochastic $p(x)$-Laplace Equations
Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）；School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）；Intelligent Game and Decision Lab, Academy of Military Sciences（军事科学院智能博弈与决策实验室）
Martingales in self-similar growth-fragmentations and their connections with random planar maps
本工作的目的是双重的。首先，我们开发的一般自相似增长破碎过程的理论，专注于自然出现在这个设置和重铸经典分支随机游动的结果在这个框架中的鞅。特别是，我们建立了多对一的增长破碎公式，并定义了增长破碎的固有区域的概念。其次，我们确定了一个与稳定Lévy过程密切相关的杰出的增长-破碎族，然后证明了这是在某些大度数随机平面映射（粗略地说，是Le Gall & Miermont稳定映射的对偶映射）的马尔可夫探索中作为周长过程的标度极限而出现的。由于这一结果，我们能够确定这些独特的生长-破碎的内在面积的规律。这概括了大玻尔兹曼三角剖分和一定的增长-碎裂过程之间的几何联系，这是在arXiv：1507.02265中建立的。
最近的预印本（arXiv：1402.2632）介绍了平面映射的增长过程，其几乎必然极限是“一致无限3连通平面映射”。Brooks，Smith，Stone and Tutte（1940）的一个经典构造将矩形的平方（即矩形被正方形平铺）与任何有限的、具有非分离根边的边根平面映射相关联。我们使用这种结构与地图增长的过程来定义一个不断增长的矩形平方序列。我们证明了序列的平方收敛到一个几乎必然的限制：一个随机无限平方的有限矩形。这提供了一致无限3-连通平面映射的一个标准平面嵌入。我们还表明，限制随机平方几乎一定有一个唯一的聚点。
