Exact Large Deviation Function in the Asymmetric Exclusion Process

By an extension of the Bethe ansatz method used by Gwa and Spohn, we obtain an exact expression for the large deviation function of the time averaged current for the fully asymmetric exclusion process in a ring containing $N$ sites and $p$ particles. Using this expression we easily recover the exact diffusion constant obtained earlier and calculate as well some higher cumulants. The distribution of the deviation $y$ of the average current is, in the limit $N\to \mathrm{\infty }$, skew and decays like $\exp -(A{y}^{5\mathrm{/}2})$ for $y \to

• \infty$and$\exp - (A’ |y|^{3/2})$for$y \to -\infty$. Surprisingly, the large deviation function has an expression very similar to the pressure (as a function of the density) of an ideal Bose or Fermi gas in $3d$.