The Asymmetric Simple Exclusion Process with Multiple Shocks

We consider the one dimensional totally asymmetric simple exclusion process with initial product distribution with densities $0 \leq \rho_0 < \rho_1 <...< \rho_n \leq 1$ in $(-\infty,c_1\ve^{-1})$, $[c_1\ve^{-1},c_2\epsilon^{-1}),...,[c_n \ve^{-1}, + \infty)$, respectively. The initial distribution has shocks (discontinuities) at $\epsilon^{-1}c_k$, k=1,...,n and we assume that in the corresponding macroscopic Burgers equation the n shocks meet in $r^*$ at time $t^*$. The microscopic position of the shocks is represented by second class particles whose distribution in the scale $\epsilon^{-1/2}$ is shown to converge to a function of n independent Gaussian random variables representing the fluctuations of these particles ``just before the meeting''. We show that the density field at time $\ve^{-1}t^*$, in the scale $\ve^{-1/2}$ and as seen from $\ve^{-1}r^*$ converges weakly to a random measure with piecewise constant density as $\ve \to 0$; the points of discontinuity depend on these limiting Gaussian variables. As a corollary we show that, as $\epsilon\to 0$, the distribution of the process at site $\epsilon^{-1}r^*+\ve^{-1/2}a$ at time $\epsilon^{-1}t^*$ tends to a non trivial convex combination of the product measures with densities $\rho_k$, the weights of the combination being explicitly computable.