Fluctuations for stationary \(q\)-TASEP. (English) Zbl 1416.82030
Summary: We consider the \(q\)-totally asymmetric simple exclusion process (\(q\)-TASEP) in the stationary regime and study the fluctuation of the position of a particle. We first observe that the problem can be studied as a limiting case of an \(N\)-particle \(q\)-TASEP with a random initial condition and with particle dependent hopping rate. Then we explain how this \(N\)-particle \(q\)-TASEP can be encoded in a dynamics on a two-sided Gelfand-Tsetlin cone described by a two-sided \(q\)-Whittaker process and present a Fredholm determinant formula for the \(q\)-Laplace transform of the position of a particle. Two main ingredients in its derivation is the Ramanujan’s bilateral summation formula and the Cauchy determinant identity for the theta function with an extra parameter. Based on this we establish that the position of a particle obeys the universal stationary KPZ distribution (the Baik-Rains distribution) in the long time limit.
MSC:
| 82C22 | Interacting particle systems in time-dependent statistical mechanics |
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 33D52 | Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) |
| 44A12 | Radon transform |
Keywords:
asymmetric simple exclusion process; Kardar-Parisi-Zhang universality; integrable probability; Macdonald processes; Frobenius determinantReferences:
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