## Reverse juggling processes. (arXiv:1706.03956v1 [math.PR])

In a recent paper, Knutson (arXiv:1601.06391) studied a Markov chain on
semi-infinite matrices $b\times \mathbb N$ over $GF(q)$ leading to two models
of reverse juggling. These transitions were the same as the time-reversed
transitions of previously studied (forward) juggling chains by Ayyer, Bouttier,
Corteel and Nunzi (Elec. J. Prob, Vol. 20, 2015) and by Ayyer, Bouttier,
Corteel, Linusson and Nunzi (arXiv:1504.02688). In this paper we generalize the
reverse juggling chains of Knutson for both single and multiple species. We
show that there are natural ways to place generic weights on the transitions
and still obtain chains where the stationary distribution have a simple form,
both for finite and infinite states. In the finite single species case, we find
the phenomenon of ultrafast convergence to the stationary distribution. In the
finite multispecies case, the stationary distribution turns out to be a
multivariate generalisation of the inversion polynomial. Lastly, we observe a
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