Random walks and contracting elements II: Translation length and quasi-isometric embedding

Continuing from Choi (2022), we study random walks on metric spaces with contracting elements. We prove that random subgroups of the isometry group of a metric space are quasi-isometrically embedded into the space. We discuss this problem in two ways, namely in the sense of random walks and counting problem. We also establish the genericity of contracting elements and the central limit theorem and its converse for translation length.