Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 056, 14 pages      arXiv:1703.07731      https://doi.org/10.3842/SIGMA.2017.056
Contribution to the Special Issue on Recent Advances in Quantum Integrable Systems

Zero Range Process and Multi-Dimensional Random Walks

Nicolay M. Bogoliubov ab and Cyril Malyshev ab
a) St.-Petersburg Department of Steklov Institute of Mathematics of RAS, Fontanka 27, St.-Petersburg, Russia
b) ITMO University, Kronverksky 49, St.-Petersburg, Russia

Received March 28, 2017, in final form July 14, 2017; Published online July 22, 2017

Abstract
The special limit of the totally asymmetric zero range process of the low-dimensional non-equilibrium statistical mechanics described by the non-Hermitian Hamiltonian is considered. The calculation of the conditional probabilities of the model are based on the algebraic Bethe ansatz approach. We demonstrate that the conditional probabilities may be considered as the generating functions of the random multi-dimensional lattice walks bounded by a hyperplane. This type of walks we call the walks over the multi-dimensional simplicial lattices. The answers for the conditional probability and for the number of random walks in the multi-dimensional simplicial lattice are expressed through the symmetric functions.

Key words: zero range process; conditional probability; multi-dimensional random walk; correlation function; symmetric functions.

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