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

TL;DR

The paper studies two natural statistics of TASEP with step initial data: the total number of jumps ${P_t}$ up to time ${t}$ and the number of active particles ${a_t}$ at time ${t}$. Using Rost's asymptotics and the Poisson behavior of the rightmost particle, it proves ${\mathbb E}P_t \sim t^{2}/6$ and ${\mathbb E}a_t \sim (1/3) t$, and establishes a direct link by showing the derivative relation ${g'(t) = {\mathbb E}a_t}$ where ${g(t)={\mathbb E}P_t}$. The results illuminate how macroscopic evolution rates relate to microscopic activity and provide a clear, elementary pathway connecting these fundamental process descriptors within the KPZ-universal class. The analysis hinges on area interpretations, Rost’s limiting profiles ${h(u)}$ and ${f(u)}$, and the Poisson dynamics of the rightmost particle, offering insights into the temporal structure of TASEP dynamics beyond height-function fluctuations.

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.

Figures (1)

• Figure 1: Graphs of $S(k,0)$ (blue) and $S(k,t)$ (black) after eight jumps: four by the first particle, three by the second, and one by the third. The area of the differences $S(k,t)-S(k,0)$, $k\in \Bbb Z$, is the number of jumps by the process up to time $t$.

Theorems & Definitions (8)

• Lemma 2.1
• Theorem 3.1
• Theorem 4.1
• Theorem 5.1
• proof
• Proposition 6.1
• Proposition 6.2
• proof