Dynamics of the leftmost particle in heterogeneous semi-infinite exclusion systems

We study the behaviour of the leftmost particle in a semi-infinite particle system on $\mathbb{Z}$, where each particle performs a continuous-time nearest-neighbour random walk, with particle-specific jump rates, subject to the exclusion interaction (i.e., no more than one particle per site). We give conditions, in terms of the jump rates on the system, under which the leftmost particle is recurrent or transient, and develop tools to study its rate of escape in the transient case, including by comparison with an $M/G/\infty$ queue. In particular we show examples in which the leftmost particle can be null recurrent, positive recurrent, ballistically transient, or subdiffusively transient. Finally we indicate the role of the initial condition in determining the dynamics, and show, for example, that sub-ballistic transience can occur started from close-packed initial configurations but not from stationary initial conditions.

💡 Research Summary

The paper investigates the long‑time behavior of the leftmost particle in a semi‑infinite exclusion process on ℤ where each particle k performs a continuous‑time nearest‑neighbour random walk with its own left and right jump rates a_k and b_k (both strictly positive and uniformly bounded). Because of the exclusion rule the ordering of particles is preserved; the state of the system can be described by the position X₁(t) of the leftmost particle together with the inter‑particle gaps η_k(t)=X_{k+1}(t)−X_k(t)−1. The gap process η(t) is shown to be an infinite Jackson network of queues: each gap η_k is the number of customers in queue k, and a particle jump corresponds to a service completion that moves a customer to a neighboring queue (or creates/removes a customer at the leftmost queue when the leftmost particle jumps).

A central analytical tool is the “customer random walk” ζ(t) on ℕ, which moves right from k to k+1 at rate a_{k+1} and left from k to k−1 at rate b_k. ζ represents the trajectory of a priority customer through the queueing network and, crucially, its recurrence or transience determines the macroscopic drift of X₁. The stationary traffic equations
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