Current Fluctuations in the Asymmetric Simple Exclusion Process and Bethe Ansatz - Part II

In this second part, after recalling briefly how the Bethe Ansatz is used for the calculation of the current fluctuations in the asymmetric simple exclusion process on a ring, I will show that the Bethe equations can be conveniently rewritten as a functional equation. This functional equation can be solved exactly in the totally asymmetric case, giving access to the complete fluctuations of the current. For the general case of a partial asymmetry, a complete solution is not known: only the three first cumulants of the current have been calculated. I will discuss the two interesting scaling limits for which these cumulants take simple expressions. These two scaling limits can be understood by simple physical arguments.