Expected number of jumps and the number of active particles in TASEP

Abstract

For a TASEP on $\mathbb Z$ with the step initial condition we identify limits as $t\to\infty$ of the expected total number of jumps until time $t>0$ and the expected number of active particles at a time $t$. We also connect the two quantities proving that non-asymptotically, that is as a function of $t>0$, the latter is the derivative of the former. Our approach builds on asymptotics derived by Rost and intensive use of the fact that the rightmost particle evolves according to the Poisson process.