Transition probability and total crossing events in the multi-species asymmetric exclusion process. (English) Zbl 1512.60064
Summary: We present explicit formulas for total crossing events in the multi-species asymmetric exclusion process (\(r\)-ASEP) with underlying \(U(\widehat{\mathfrak{sl}}_{r+1})\) 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 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.
MSC:
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 82B23 | Exactly solvable models; Bethe ansatz |
| 05E05 | Symmetric functions and generalizations |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 82D30 | Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses) |
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