Discrete geometric analysis of the scaling limit and its applications

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We studied the random walks on groups and the geometry of random graphs having their origins in statistical physics. Those are related to the central theme of this project. In particular, we focused on the important example in the three-dimensional solvable Lie groups and its discrete subgroups, and showed that the harmonic measure associated with the random walk drastically changes its measure-theoretic property according to the step distribution. We also studied the related problems arising from this subject.Furthermore, we estimated the isoperimetric constant which has an important role in the geometry of the graphs for locally free product of random graphs.