论文标题
相对双曲组中的本地限制定理I：粗略估计
Local limit theorems in relatively hyperbolic groups I : rough estimates
论文作者
论文摘要
这是两篇论文中的第一篇，该论文涉及相对双曲线组中局部极限定理的第一篇论文。在第一篇论文中，我们证明了绿色功能的粗略估计。在此过程中，我们介绍了相对自动化的概念，该概念在两篇论文中都会有用，并表明相对双曲线相对自动。我们还定义了在相对透明的群体上随机步行的光谱正差的概念。然后，我们使用绿色功能的估计值证明$ p_n \ asymp r^n n^{ - 3/2} $用于频谱积极反复的随机步行，其中$ p_n $是在时间n返回原点的可能性，r是随机步行的频谱半径。
This is the first of a series of two papers dealing with local limit theorems in relatively hyperbolic groups. In this first paper, we prove rough estimates for the Green function. Along the way, we introduce the notion of relative automaticity which will be useful in both papers and we show that relatively hyperbolic groups are relatively automatic. We also define the notion of spectral positive-recurrence for random walks on relatively hy-perbolic groups. We then use our estimates for the Green function to prove that $p_n \asymp R^n n^{-3/2}$ for spectrally positive-recurrent random walks, where $p_n$ is the probability of going back to the origin at time n and where R is the spectral radius of the random walk.
