Abstract

We present explicit formulas for total crossing events in the multi-species asymmetric exclusion process (r-ASEP) with underlying symmetry. In the case of the two-species TASEP these can be derived using an explicit expression for the general transition probability on 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 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.

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