Orthogonal Polynomial Stochastic Duality Functions for Multi-Species SEP$(2j)$ and Multi-Species IRW

Zhengye Zhou

Symmetry, integrability and geometry: methods and applications, Tome 17 (2021)

Résumé

We obtain orthogonal polynomial self-duality functions for multi-species version of the symmetric exclusion process (SEP$(2j)$) and the independent random walker process (IRW) on a finite undirected graph. In each process, we have $n>1$ species of particles. In addition, we allow up to $2j$ particles to occupy each site in the multi-species SEP$(2j)$. The duality functions for the multi-species SEP$(2j)$ and the multi-species IRW come from unitary intertwiners between different $*$-representations of the special linear Lie algebra $\mathfrak{sl}_{n+1}$ and the Heisenberg Lie algebra $\mathfrak{h}_n$, respectively. The analysis leads to multivariate Krawtchouk polynomials as orthogonal duality functions for the multi-species SEP$(2j)$ and homogeneous products of Charlier polynomials as orthogonal duality functions for the multi-species IRW.

Détail

Keywords: orthogonal duality, multi-species SEP$(2j)$, multi-species IRW.

Bibliographie

[1] Ayala M., Carinci G., Redig F., “Quantitative Boltzmann–Gibbs principles via orthogonal polynomial duality”, J. Stat. Phys., 171 (2018), 980–999, arXiv: 1712.08492 | DOI | MR | Zbl

[2] Ayala M., Carinci G., Redig F., “Higher order fluctuation fields and orthogonal duality polynomials”, Electron. J. Probab., 26 (2021), 27, 35 pp., arXiv: 2004.08412 | DOI | MR | Zbl

[3] Borodin A., Corwin I., Sasamoto T., “From duality to determinants for $q$-TASEP and ASEP”, Ann. Probab., 42 (2014), 2314–2382, arXiv: 1207.5035 | DOI | MR | Zbl

[4] Caputo P., “On the spectral gap of the Kac walk and other binary collision processes”, ALEA Lat. Am. J. Probab. Math. Stat., 4 (2008), 205–222, arXiv: 0807.3415 | MR | Zbl

[5] Carinci G., Franceschini C., Giardinà C., Groenevelt W., Redig F., “Orthogonal dualities of Markov processes and unitary symmetries”, SIGMA, 15 (2019), 053, 27 pp., arXiv: 1812.08553 | DOI | MR | Zbl

[6] Chen J. P., Sau F., “Higher-order hydrodynamics and equilibrium fluctuations of interacting particle systems”, Markov Process. Related Fields, 27 (2021), 339–380, arXiv: 2008.13403 | Zbl

[7] Corwin I., Shen H., Tsai L.-C., “${\rm ASEP}(q,j)$ converges to the KPZ equation”, Ann. Inst. Henri Poincaré Probab. Stat., 54 (2018), 995–1012, arXiv: 1602.01908 | DOI | MR | Zbl

[8] Dermoune A., Heinrich P., “Spectral gap for multicolor nearest-neighbor exclusion processes with site disorder”, J. Stat. Phys., 131 (2008), 117–125 | DOI | MR | Zbl

[9] Franceschini C., Giardinà C., Stochastic duality and orthogonal polynomials, arXiv: 1701.09115

[10] Giardinà C., Kurchan J., Redig F., Vafayi K., “Duality and hidden symmetries in interacting particle systems”, J. Stat. Phys., 135 (2009), 25–55, arXiv: 0810.1202 | DOI | MR | Zbl

[11] Griffiths R. C., “Orthogonal polynomials on the multinomial distribution”, Aust. J. Stat., 13 (1971), 27–35 | DOI | MR | Zbl

[12] Groenevelt W., “Orthogonal stochastic duality functions from Lie algebra representations”, J. Stat. Phys., 174 (2019), 97–119, arXiv: 1709.05997 | DOI | MR | Zbl

[13] Iliev P., “A Lie-theoretic interpretation of multivariate hypergeometric polynomials”, Compos. Math., 148 (2012), 991–1002, arXiv: 1101.1683 | DOI | MR | Zbl

[14] Kuan J., “A multi-species ${\rm ASEP}(q,j)$ and $q$-TAZRP with stochastic duality”, Int. Math. Res. Not., 2018 (2018), 5378–5416, arXiv: 1605.00691 | DOI | MR | Zbl

[15] Kuan J., Stochastic fusion of interacting particle systems and duality functions, arXiv: 1908.02359

[16] Redig F., Sau F., “Factorized duality, stationary product measures and generating functions”, J. Stat. Phys., 172 (2018), 980–1008, arXiv: 1702.07237 | DOI | MR | Zbl

[17] Zhou Z., “Hydrodynamic limit for a $d$-dimensional open symmetric exclusion process”, Electron. Commun. Probab., 25 (2020), 76, 8 pp., arXiv: 2004.14279 | DOI | MR | Zbl