This paper is concerned with Markov chain models for flows of a finite population among a set of groups, where the individuals base their decisions on which group to go next partially on the current frequency distribution (profile). For a certain class of these models, the transition matrix of the profile process is analyzed algebraically, leading to surprisingly simple asymptotic results. Furthermore, in a model with after effects, the absorption probabilities are derived.
"Asymptotic behavior of some interactive population flow models." Ann. Appl. Probab. 7 (4) 837 - 854, November 1997. https://doi.org/10.1214/aoap/1043862413