each site x in z is initially occupied by a particle of color -x. across each bond (x,x+1) particles swap places at rate 1 or q<1 depending on whether they are in reverse order (e.g. color 2 then 1) or order (color 1 then 2). this process describes a bijection of z-->z which starts maximally in reverse order and randomly drifts towards being ordered. another name for this model is the "colored asymmetric simple exclusion process". i will explain how to use the yang-baxter equation along with techniques involving gibbs measures to extract the space-time scaling limit of this process, as well as a discrete time analog known as the "stochastic six vertex model". the limit is described by objects in the kardar-parisi-zhang universality class, namely the airy sheet, directed landscape and kpz fixed point. this is joint work with amol aggarwal and milind hegde.