Abstract: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.
| Comments: | 24 pages, 1 figure; v2: minor revision |
| Subjects: | Probability (math.PR) |
| MSC classes: | 60K35 (Primary), 60J27, 60K25, 90B22 (Secondary) |
| Cite as: | arXiv:2602.03492 [math.PR] |
| (or arXiv:2602.03492v2 [math.PR] for this version) | |
| https://doi.org/10.48550/arXiv.2602.03492 arXiv-issued DOI via DataCite |
From: Andrew R. Wade [view email]
[v1]
Tue, 3 Feb 2026 13:09:17 UTC (75 KB)
[v2]
Tue, 26 May 2026 10:54:17 UTC (75 KB)
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