Given a pair of filling multicurves on a closed surface, Thurston ('88) introduced a $\mathrm{PSL}(2, \mathbb{R})$-representation of a subgroup generated by the multitwists along the multicurves by thinking of the subgroup as a stabilizer subgroup (in the Mapping class group) of a hyperbolic disk isometrically embedded in the Teichmüller space. In most cases, such a stabilizer subgroup is free of rank two. We consider a random walk on the stabilizer subgroup and study asymptotic behaviors of the random walk. This talk discusses the following two main theorems, which were known for much more general settings but under finite support conditions by Joseph Maher ('11, '12), Joseph Maher and Giulio Tiozzo ('18), and François Dahmani and Camille Horbez ('18):

- Strong law of large numbers of topological entropy along the random walk under finite first-moment conditions.

- Almost every sample path consists of all but finitely many pseudo-Anosov mapping classes without any assumption on the moment.

This is based on the joint work with Hyungryul Baik and Inhyeok Choi. While the same results for much more general settings were also obtained in the sequel joint work with Hyungryul Baik and Inhyeok Choi, which was reported in Yale Geometry and Topology Seminar in 2021, the argument in the above specific situation is totally different from the general settings and is of independent interest itself.