Dynamics of the Leftmost Particle in Heterogeneous Semi-infinite Exclusion Systems

M.V. Menshikov, S.Yu. Popov, A.R. Wade

2026, v.32, Issue 2, 205-232

ABSTRACT

We study the behaviour of the leftmost particle in a semi-infinite particle system on 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 of 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/∞ 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.

doi:10.61102/1024-2953-mprf.2026.32.2.002

Keywords: Exclusion process, infinite Jackson network, interacting particle system, invariant measures, transience, null recurrence, rate of escape, subdiffusive, M/G/∞ queue

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