Local limit of labeled trees and expected volume growth in a random quadrangulation

An Improved Lower Bound on the Largest Common Subtree of Random Leaf-Labeled Binary Trees

It is known that the size of the largest common subtree (i.e., the maximum agreement subtree) of two independent random binary trees with $n$ given labeled leaves is of order between $n^{0.366}$ and $n^{1/2}$. We improve the lower bound to order $n^{0.4464}$ by constructing a common subtree recursively and by proving a lower bound for its asymptotic growth. The construction is a modification of an algorithm proposed by D. Aldous by splitting the tree at the centroid and by proceeding recursively.

Conformal growth rates and spectral geometry on distributional limits of graphs

For a unimodular random graph $(G,ρ)$, we consider deformations of its intrinsic path metric by a (random) weighting of its vertices. This leads to the notion of the conformal growth exponent of $(G,ρ)$, which is the best asymptotic degree of volume growth of balls that can be achieved by such a reweighting. Under moment conditions on the degree of the root, we show that the conformal growth exponent of a unimodular random graph bounds its almost sure spectral dimension. This has interesting consequences for many low-dimensional models. The consequences in dimension two are particularly strong…

矩形吉尔伯特镶嵌

Rectangular Gilbert Tessellation

本文提出一种矩形吉尔伯特镶嵌模型，用于近似元胞自动机，推导了线段长度分布的指数尾部界，证明相关性指数衰减，并给出边界射线数量的线性阶结果。

A non-increasing tree growth process for recursive trees and applications

We introduce a non-increasing tree growth process $((T_n,σ_n),\, n\ge 1)$, where $T_n$ is a rooted labeled tree on $n$ vertices and $σ_n$ is a permutation of the vertex labels. The construction of $(T_{n},σ_n)$ from $(T_{n-1},σ_{n-1})$ involves rewiring a random (possibly empty) subset of edges in $T_{n-1}$ towards the newly added vertex; as a consequence $T_{n-1} \not\subset T_n$ with positive probability. The key feature of the process is that the shape of $T_n$ has the same law as that of a random recursive tree, while the degree distribution of any given vertex is not monotonous in the pro…

Limit distributions for KPZ growth models with spatially homogeneous random initial conditions

For stationary KPZ growth in 1+1 dimensions the height fluctuations are governed by the Baik-Rains distribution. Using the totally asymmetric single step growth model, alias TASEP, we investigate height fluctuations for a general class of spatially homogeneous random initial conditions. We prove that for TASEP there is a one-parameter family of limit distributions, labeled by the diffusion coefficient of the initial conditions. The distributions are defined through a variational formula. We use Monte Carlo simulations to obtain their numerical plots. Also discussed is the connection to the six…

匹配基因树与物种树的祖先构型总数在两种物种树模型下的分布

The distributions under two species-tree models of the total number of ancestral configurations for matching gene trees and species trees

该研究针对基因树与物种树拓扑匹配的情形，用解析组合方法分析了均匀和Yule两种模型下随机标记拓扑的祖先构型总数的分布，证明其渐近服从对数正态分布并给出了均值与方差的渐近增长式。

Iterated foldings of discrete spaces and their limits: candidates for the role of Brownian map in higher dimensions

In this last decade, an important stochastic model emerged: the Brownian map. It is the limit of various models of random combinatorial maps after rescaling: it is a random metric space with Hausdorff dimension 4, almost surely homeomorphic to the 2-sphere, and possesses some deep connections with Liouville quantum gravity in 2D. In this paper, we present a sequence of random objects that we call $D$th-random feuilletages (denoted by ${\bf r}[D]$), indexed by a parameter $D\geq 0$ and which are candidate to play the role of the Brownian map in dimension $D$. The construction relies on some obj…