Large deviations for random walks on Gromov-hyperbolic spaces

Counting and boundary limit theorems for representations of Gromov-hyperbolic groups

Given a Gromov-hyperbolic group $G$ endowed with a finite symmetric generating set, we study the statistics of counting measures on the spheres of the associated Cayley graph under linear representations of $G$. More generally, we obtain a weak law of large numbers for subadditive functions, echoing the classical Fekete lemma. For strongly irreducible and proximal representations, we prove a counting central limit theorem with a Berry--Esseen type error rate and exponential large deviation estimates. Moreover, in the same setting, we show convergence of interpolated normalized matrix norms alo…

Gromov双曲空间与Teichmüller空间上随机等距变换平移长度的线性增长

Linear growth of translation lengths of random isometries on Gromov hyperbolic spaces and Teichmüller spaces

该研究针对Gromov双曲空间等距群上的随机游走，证明无矩条件下非初等随机游走平移长度至少线性增长，有限一阶矩下增长率等于漂移，还将技术推广到Teichmüller空间，在有限一阶矩下证明谱定理，拓展了已有结果。

Average Gromov hyperbolicity and the Parisi ansatz

Gromov hyperbolicity of a metric space measures the distance of the space from a perfect tree-like structure. The measure has a "worst-case" aspect to it, in the sense that it detects a region in the space which sees the maximum deviation from tree-like structure. In this article we introduce an "average-case" version of Gromov hyperbolicity, which detects whether the "most of the space", with respect to a given probability measure, looks like a tree. The main result of the paper is that if this average hyperbolicity is small, then the space can be approximately embedded in a tree. The proof u…

Central Limit Theorems for Gromov Hyperbolic Groups

In this paper we study asymptotic properties of symmetric and non-degenerate random walks on transient hyperbolic groups. We prove a central limit theorem and a law of iterated logarithm for the drift of a random walk, extending previous results by S. Sawyer and T. Steger and F. Ledrappier for certain CAT minus one groups. The proofs use a result by A. Ancona on the identification of the Martin boundary of a hyperbolic group with its Gromov boundary. We also give a new interpretation, in terms of Hilbert metrics, of the Green metric, first introduced by S. Brofferio and S. Blachere.

Gromov双曲空间中的重心与大数定律

Barycenters and a law of large numbers in Gromov hyperbolic spaces

本文研究Gromov双曲空间中概率测度的重心，证明收缩性质、有限均匀分布重心的确定性逼近和大数定律，将CAT(0)空间结果推广至含双曲常数附加项的情形。

双曲群上的随机游走与真幂

Random walk on hyperbolic groups and proper powers

本文研究了双曲群上的对称随机游走，证明到达真幂的概率与返回单位元的概率具有相同的指数衰减率。