The stationary distribution of a continuous-time Markov chain is generally a complicated function of its transition rates. However, if the rates are i.i.d. random variables whose common distribution satisfies certain tail conditions, then the stationary distribution is essentially a simple function of the exit rates out of each state. This is the main result of a new preprint with Frank den Hollander and Dana Randall, which generalizes and makes a precise prediction by Chvykov et al. (2021) and settles a question raised by Bordenave, Caputo, and Chafaï (2012) under certain assumptions.
You can read the preprint here.