Abstract
We discuss the conditions for unique ergodicity of a collective random walk on a continuous circle. Individual particles in this collective motion perform independent (and different in general) random walks conditioned by the assumption that the particles cannot overrun each other. In addition to sufficient conditions for unique ergodicity, we discover a new and unexpected way for its violation due to excessively large local jumps. Necessary and sufficient conditions for the unique ergodicity of the deterministic version of this system are also obtained. Technically, our approach is based on the interlacing property of the spin function which describes the states of pairs of particles in the coupled processes being studied.
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