论文标题
随机步行的漂移在有限体积均匀空间的Abelian封面上
Drift of random walks on abelian covers of finite volume homogeneous spaces
论文作者
论文摘要
令$ g $为一个连接的简单真实的谎言组，$λ_{0} \ subseteq g $ a晶格和$λ\linlhdλ_{0} $正常子组，以至于$λ_{0}/λ\ simeq \ simeq \ mathbb {z}^d $。我们研究有限体积均匀空间的$ \ mathbb {z}^d $ - cover $λ\ backslash g $的随机步行的漂移。这次步行是由$ g $上的Zariski强度紧凑型概率$μ$定义的。我们首先假设覆盖映射$λ\ backslash g \rightArrowλ_{0} \ backslash g $不会展开任何$λ_{0} \ backslash g $的cusp，并计算\ emph {emph {every}起点的漂移。然后，我们删除此假设，并几乎在任何地方描述漂移。维度2的双曲线歧管的情况以非分配类型的行为脱颖而出。在这种情况下，还表征了轨迹的复发。
Let $G$ be a connected simple real Lie group, $Λ_{0}\subseteq G$ a lattice and $Λ\unlhd Λ_{0}$ a normal subgroup such that $Λ_{0}/Λ\simeq \mathbb{Z}^d$. We study the drift of a random walk on the $\mathbb{Z}^d$-cover $Λ\backslash G$ of the finite volume homogeneous space $Λ_{0}\backslash G$. This walk is defined by a Zariski-dense compactly supported probability measure $μ$ on $G$. We first assume the covering map $Λ\backslash G\rightarrow Λ_{0}\backslash G$ does not unfold any cusp of $Λ_{0}\backslash G$ and compute the drift at \emph{every} starting point. Then we remove this assumption and describe the drift almost everywhere. The case of hyperbolic manifolds of dimension 2 stands out with non-converging type behaviors. The recurrence of the trajectories is also characterized in this context.
