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 \(U_q(\widehat{\mathfrak{sl}}_{r+1})\) symmetry. In the case of the two-species TASEP these can be derived using an explicit expression for the general transition probability on \(\mathbb{Z}\) in terms of a multiple contour integral derived from a nested Bethe ansatz approach. For the general \(r\)-ASEP we employ a vertex model approach within which the probability of total crossing can be derived from partial symmetrization 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 \(2\)-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.