‰ªŽR‘åŠwŠÂ‹«—HŠw•”Acta Medica Okayama1341-9099211997A Critical Phenomenon Appearing in the Process of Patricle Diffusion in Classical Statistical Mechanics18ENAkihitoHoraAn aspect of the cut-off phenomenon is reviewed. It is a sort of critical phenomenon observed in a wide variety of diffusion. We introduce a framework in which such a phenomenon can be understood and analyzed in a rigorous manner. We illustrate the mechanism with some concrete models.No potential conflict of interest relevant to this article was reported.‰ªŽR‘åŠwŠÂ‹«—HŠw•”Acta Medica Okayama1341-9099111996Random walks and isotropic Markov chains on homogeneous spaces2126ENAkihitoHoraLet G be a topological group acting on S transitively from the left with a compact stabilizer K. We show that every isotropic (i.e. spatially homogeneous w.r.t. the G-actions) Markov chain on S can be lifted to a right random walk on G and give a one-to-one correspondence between the isotropic Markov chains on S and the totality of sequences of probabilities (ƒË,ƒÊ1,ƒÊ2,¥¥¥) where ƒË is a probability on G/K and each ƒÊn is that on K_G/K.No potential conflict of interest relevant to this article was reported.