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Juggler's Exclusion Process

Part of: Special processes Time-dependent statistical mechanics (dynamic and nonequilibrium)

Published online by Cambridge University Press:  04 February 2016

Lasse Leskelä  and
Harri Varpanen
Lasse Leskelä*
Affiliation:
University of Jyväskylä
Harri Varpanen*
Affiliation:
Aalto University
*
∗ Postal address: Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35, 40014, Finland. Email address: lasse.leskela@iki.fi
∗∗ Postal address: Department of Mathematics and Systems Analysis, Aalto University, PO Box 11100, 00076 Aalto, Finland. Email address: harri.varpanen@tkk.fi
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Abstract

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Juggler's exclusion process describes a system of particles on the positive integers where particles drift down to zero at unit speed. After a particle hits zero, it jumps into a randomly chosen unoccupied site. We model the system as a set-valued Markov process and show that the process is ergodic if the family of jump height distributions is uniformly integrable. In a special case where the particles jump according to a set-avoiding memoryless distribution, the process reaches its equilibrium in finite nonrandom time, and the equilibrium distribution can be represented as a Gibbs measure conforming to a linear gravitational potential.

Keywords

Exclusion processjuggling patternset-valued Markov processergodicitypositive recurrenceset-avoiding memoryless distributionnoncolliding random walkGibbs measuremaximum entropy

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 82C41: Dynamics of random walks, random surfaces, lattice animals, etc.
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 1 , March 2012 , pp. 266 - 279
Copyright
© Applied Probability Trust 

References

Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York Google Scholar
Cover, T. M. and Thomas, J. A. (2006). Elements of Information Theory, 2nd edn. Wiley-Interscience, Hoboken, NJ.Google Scholar
Grimmett, G. (2010). Probability on Graphs. Cambridge University Press.Google Scholar
Kelly, F. P. (1979). Reversibility and Stochastic Networks. John Wiley, Chichester.Google Scholar
Knutson, A., Lam, T. and Speyer, D. E. (2011). Positroid varieties: Juggling and geometry. Preprint. Available at http://arxiv.org/abs/1111.3660v1.Google Scholar
Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York.Google Scholar
Martin, J. and Schmidt, P. (2011). Multi-type TASEP in discrete time. ALEA 8, 303–333.Google Scholar
Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.Google Scholar
Stanley, R. P. (1997). Enumerative Combinatorics, Vol. 1. Cambridge University Press.Google Scholar
Warrington, G. S. (2005). Juggling probabilities. Amer. Math. Monthly 112, 105–118.Google Scholar