Geodesic
Zero Range Process and Multi-Dimensional Random Walks
Symmetry, integrability and geometry: methods and applications, Tome 13 (2017) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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.
Keywords: zero range process; conditional probability; multi-dimensional random walk; correlation function; symmetric functions. 
@article{SIGMA_2017_13_a55,
     author = {Nicolay M. Bogoliubov and Cyril Malyshev},
     title = {Zero {Range} {Process} and {Multi-Dimensional} {Random} {Walks}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2017},
     volume = {13},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a55/}
}
TY  - JOUR
AU  - Nicolay M. Bogoliubov
AU  - Cyril Malyshev
TI  - Zero Range Process and Multi-Dimensional Random Walks
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2017
VL  - 13
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a55/
LA  - en
ID  - SIGMA_2017_13_a55
ER  - 
%0 Journal Article
%A Nicolay M. Bogoliubov
%A Cyril Malyshev
%T Zero Range Process and Multi-Dimensional Random Walks
%J Symmetry, integrability and geometry: methods and applications
%D 2017
%V 13
%U http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a55/
%G en
%F SIGMA_2017_13_a55
Nicolay M. Bogoliubov; Cyril Malyshev. Zero Range Process and Multi-Dimensional Random Walks. Symmetry, integrability and geometry: methods and applications, Tome 13 (2017). http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a55/

[1] Bogoliubov N. M., “Boxed plane partitions as an exactly solvable boson model”, J. Phys. A: Math. Gen., 38 (2005), 9415–9430, arXiv: cond-mat/0503748 | DOI | MR | Zbl

[2] Bogoliubov N. M., “The $XXO$ {H}eisenberg chain and random walks”, J. Math. Sci., 138 (2006), 5636–5643 | DOI | MR

[3] Bogoliubov N. M., “Calculation of correlation functions in totally asymmetric exactly solvable models on a ring”, Theoret. and Math. Phys., 175 (2013), 755–762 | DOI | MR | Zbl

[4] Bogoliubov N. M., Quantum walks and phase operators, in preparation

[5] Bogoliubov N. M., Bullough R. K., Pang G. D., “Exact solution of the $q$-boson hopping model”, Phys. Rev. B, 47 (1993), 11495–11498 | DOI | MR

[6] Bogoliubov N. M., Malyshev C., “Correlation functions of $XX0$ Heisenberg chain, $q$-binomial determinants, and random walks”, Nuclear Phys. B, 879 (2014), 268–291, arXiv: 1401.7624 | DOI | MR | Zbl

[7] Bogoliubov N. M., Malyshev C. L., “Integrable models and combinatorics”, Russ. Math. Surv., 70 (2015), 789–856 | DOI | MR | Zbl

[8] Bogoliubov N. M., Nassar T., “On the spectrum of the non-Hermitian phase-difference model”, Phys. Lett. A, 234 (1997), 345–350 | DOI | MR | Zbl

[9] Brak R., Essam J. W., “Simple asymmetric exclusion model and lattice paths: bijections and involutions”, J. Phys. A: Math. Theor., 45 (2012), 494007, 22 pp., arXiv: 1209.1446 | DOI | MR | Zbl

[10] Bravyi S., Caha L., Movassagh R., Nagaj D., Shor P. W., “Criticality without frustration for quantum spin-1 chains”, Phys. Rev. Lett., 109 (2012), 207202, 5 pp., arXiv: 1203.5801 | DOI

[11] Carruters P., Nieto M., “Phase and angle variables in quantum mechanics”, Rev. Modern Phys., 40 (1968), 411–440 | DOI

[12] Evans M. R., Hanney T., “Nonequilibrium statistical mechanics of the zero-range process and related models”, J. Phys. A: Math. Gen., 38 (2005), R195–R240, arXiv: cond-mat/0501338 | DOI | MR | Zbl

[13] Faddeev L. D., “Quantum completely integrable models in field theory”, Sov. Sci. Rev. Sect. C, Math. Phys. Rev., 1 (1980), 107–155 | MR | Zbl

[14] Forrester P. J., “Random walks and random permutations”, J. Phys. A: Math. Gen., 34 (2001), L417–L423, arXiv: math.CO/9907037 | DOI | MR | Zbl

[15] Großkinsky S., Schütz G. M., Spohn H., “Condensation in the zero range process: stationary and dynamical properties”, J. Stat. Phys., 113 (2003), 389–410, arXiv: cond-mat/0302079 | DOI | MR | Zbl

[16] Kanai M., “Exact solution of the zero-range process: fundamental diagram of the corresponding exclusion process”, J. Phys. A: Math. Theor., 40 (2007), 7127–7138, arXiv: cond-mat/0701190 | DOI | MR | Zbl

[17] Korepin V. E., Bogoliubov N. M., Izergin A. G., Quantum inverse scattering method and correlation functions, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1993 | DOI | MR | Zbl

[18] Krattenthaler C., “Lattice path enumeration”, Handbook of Enumerative Combinatorics, Discrete Math. Appl. (Boca Raton), ed. M. Bóna, CRC Press, Boca Raton, FL, 2015, 589–678, arXiv: 1503.05930 | MR | Zbl

[19] Kulish P. P., Damaskinsky E. V., “On the {$q$} oscillator and the quantum algebra ${\rm su}_q(1,1)$”, J. Phys. A: Math. Gen., 23 (1990), L415–L419 | DOI | MR | Zbl

[20] Kulish P. P., Sklyanin E. K., “Quantum spectral transform method. Recent developments”, Integrable Quantum Field Theories, Lecture Notes in Phys., 151, Springer, Berlin–New York, 1982, 61–119 | DOI | MR

[21] Kuniba A., Maruyama S., Okado M., “Inhomogeneous generalization of a multispecies totally asymmetric zero range process”, J. Stat. Phys., 164 (2016), 952–968, arXiv: 1602.00764 | DOI | MR | Zbl

[22] Macdonald I. G., Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, Oxford Science Publications, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1995 | MR

[23] Mackay T. D., Bartlett S. D., Stephenson L. T., Sanders B. C., “Quantum walks in higher dimensions”, J. Phys. A: Math. Gen., 35 (2002), 2745–2753, arXiv: quant-ph/0108004 | DOI | MR | Zbl

[24] Mortimer P. R. G., Prellberg T., “On the number of walks in a triangular domain”, Electron. J. Combin., 22 (2015), 1.64, 15 pp., arXiv: 1402.4448 | MR | Zbl

[25] Povolotsky A. M., “Bethe ansatz solution of zero-range process with nonuniform stationary state”, Phys. Rev. E, 69 (2004), 061109, 7 pp., arXiv: cond-mat/0401249 | DOI | MR

[26] Povolotsky A. M., “On the integrability of zero-range chipping models with factorized steady states”, J. Phys. A: Math. Theor., 46 (2013), 465205, 25 pp., arXiv: 1308.3250 | DOI | MR | Zbl

[27] Povolotsky A. M., Mendes J. F. F., “Bethe ansatz solution of discrete time stochastic processes with fully parallel update”, J. Stat. Phys., 123 (2006), 125–166, arXiv: cond-mat/0411558 | DOI | MR | Zbl

[28] Romanelli A., Donangelo R., Portugal R., Marquezino F. L., “Thermodynamics of $N$-dimensional quantum walks”, Phys. Rev. A, 90 (2014), 022329, 9 pp., arXiv: 1408.5300 | DOI

[29] Salberger O., Korepin V., Fredkin spin chain, arXiv: 1605.03842

[30] Spitzer F., “Interaction of Markov processes”, Adv. Math., 5 (1970), 246–290 | DOI | MR | Zbl

[31] Stanley R. P., Enumerative combinatorics, v. 1, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, Cambridge, 1996 | DOI | MR

[32] Stanley R. P., Enumerative combinatorics, v. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999 | DOI | MR | Zbl