The density of the ISE and local limit laws for embedded trees

Random Planar Lattices and Integrated SuperBrownian Excursion

In this paper, a surprising connection is described between a specific brand of random lattices, namely planar quadrangulations, and Aldous' Integrated SuperBrownian Excursion (ISE). As a consequence, the radius r_n of a random quadrangulation with n faces is shown to converge, up to scaling, to the width r=R-L of the support of the one-dimensional ISE. More generally the distribution of distances to a random vertex in a random quadrangulation is described in its scaled limit by the random measure ISE shifted to set the minimum of its support in zero. The first combinatorial ingredient is an e…

基于独立随机变量排序的排列

Permutations from ranking independent random variables

本文证实了关于独立随机变量排序所得排列分布半代数维数的猜想，给出上界的另一证明，并回答了Ed Crane关于Mallows分布无法作为$n\geq4$时排列分布的问题

Classes of Skorokhod Embeddings for the Simple Symmetric Random Walk

The Skorokhod Embedding problem is well understood when the underlying process is a Brownian motion. We examine the problem when the underlying is the simple symmetric random walk and when no external randomisation is allowed. We prove that any measure on Z can be embedded by means of a minimal stopping time. However, in sharp contrast to the Brownian setting, we show that the set of measures which can be embedded in a uniformly integrable way is strictly smaller then the set of centered probability measures: specifically it is a fractal set which we characterise as an iterated function system…

Gromov-Wasserstein对齐的极限定律及其在图同构检验中的应用

Limit Laws for Gromov-Wasserstein Alignment with Applications to Testing Graph Isomorphisms

本文首次在熵正则化GW距离下推导了离散、半离散及矩约束一般分布的GW经验距离极限定律，并基于此提出无需重采样的高效估计方法，应用于无标签图同构分布检验。