# The simple exclusion process with multiple shocks

### P.A. Ferrari, L.R.G. Fontes, M.E. Vares

We consider the one dimensional totally asymmetric simple exclusion process with initial product distribution with densities $0\leq\rho<\be<\la\leq 1$ in $(-\infty,0]$, $(0,\varepsilon^{-1}]$ and $(\varepsilon^{-1}, +\infty)$ respectively. The initial distribution has shocks (discontinuities) at $0$ and $\varepsilon^{-1}$. In the corresponding macroscopic Burgers equation the two shocks meet in $r^*$ at time $t^*$. We look at the distribution of the process at site $\varepsilon^{-1}r^*$ at time $\varepsilon^{-1}t^*$ and show that as $\varepsilon\to 0$ the distribution goes to a non trivial convex combination of the product measures with densities $\rho$, $\be$ and $\la$. We compute explicitly the weights of the combination. The proof is based on a study of the fluctuations of the microscopic shocks.