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Current fluctuations of the stationary ASEP and six-vertex model. (English) Zbl 1403.60081

In this paper, the author considers an asymmetric simple exclusion process (ASEP). Note that the ASEP is a particle system on \(\mathbb Z\), such that at most one particle occupies any site. Particles may jump to the left with exponential rate \(L\); jump to the right with exponential rate \(R\) and the ones that jump to occupied locations are suppressed and assumed that \(R>L\). The ASEP starts with the stationary initial data, i.e., from the Bernoulli product measure: each site of \(\mathbb Z\) is occupied independently with a given probability. By the condition \(R>L\), the particles in the ASEP on average jump in the right direction. This jumping is captured by a natural statistic called the current. The author is interested in the behavior of the particle system, through the behavior of its current, when time \(T\to +\infty\). The KPZ (Kardar-Parisi-Zhang) universality predicts that the fluctuations of the current of the ASEP are on scale \(T^{1/3}\), and moreover the distribution of these fluctuations becomes the Baik-Rains distribution as \(T\to +\infty\). The author proves these universality predictions for the ASEP. Moreover, the author is interested to integrability and fluctuations of the translation invariant stochastic six-vertex model. This is discrete time model which is related to the ASEP. A relation between translation invariant measures for the stochastic six-vertex model to Gibbs measures of the symmetric six-vertex model with arbitrary vertex weights is established.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C22 Interacting particle systems in time-dependent statistical mechanics
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