Transition probability and total crossing events in the multi-species asymmetric exclusion process

We present explicit formulas for total crossing events in the multi-species asymmetric exclusion process (r-ASEP) with underlying U-q(sl(r+1)) Z r-ASEP we employ a vertex model approach within which the probability of total crossing can be derived from partial symmetrisation of an explicit high rank rainbow partition function. In the case of r-TASEP, the total crossing probability can be show to reduce to a multiple integral over the product of r determinants. For two-TASEP we additionally derive convenient formulas for cumulative total crossing probabilities using Bernoulli-step initial conditions for particles of type 2 and type 1 respectively.