The rotation number integer quantization effect in groups acting on the circle

V.M. Buchstaber, O.V. Karpov, and S.I. Tertychnyi initiated the study of the rotation number integer quantization effect for a class of dynamical systems on a torus that includes dynamical systems modeling the dynamics of the Josephson junction. Based on this approach, we study the rotation number integer quantization effect in Artin braid groups and other finitely generated groups acting on the circle. In this case, we find the following manifestation of the quantization effect. Assume that a finitely generated group G acts proximally on the circle by orientation-preserving homeomorphisms. Then for almost every path of any non-degenerate random walk on G, the proportion of elements with integer rotation number in the initial section of the path tends to 1 as the length of the section approaches infinity.