## 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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### References:

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