Vertex electrical model: Lagrangian and nonnegativity properties
Teoretičeskaâ i matematičeskaâ fizika, Tome 210 (2022) no. 2, pp. 250-258 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We construct an embedding of the space of electrical networks to the totally nonnegative Lagrangian Grassmannian in a generic situation with the help of the technique of vertex integrable models of statistical mechanics.
Keywords: electrical networks, vertex model, total positivity.
@article{TMF_2022_210_2_a4,
     author = {D. V. Talalaev},
     title = {Vertex electrical model: {Lagrangian} and nonnegativity properties},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {250--258},
     year = {2022},
     volume = {210},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2022_210_2_a4/}
}
TY  - JOUR
AU  - D. V. Talalaev
TI  - Vertex electrical model: Lagrangian and nonnegativity properties
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2022
SP  - 250
EP  - 258
VL  - 210
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2022_210_2_a4/
LA  - ru
ID  - TMF_2022_210_2_a4
ER  - 
%0 Journal Article
%A D. V. Talalaev
%T Vertex electrical model: Lagrangian and nonnegativity properties
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2022
%P 250-258
%V 210
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2022_210_2_a4/
%G ru
%F TMF_2022_210_2_a4
D. V. Talalaev. Vertex electrical model: Lagrangian and nonnegativity properties. Teoretičeskaâ i matematičeskaâ fizika, Tome 210 (2022) no. 2, pp. 250-258. http://geodesic.mathdoc.fr/item/TMF_2022_210_2_a4/

[1] A. Postnikov, Total positivity, Grassmannians, and networks, arXiv: math/0609764

[2] P. Galashin, P. Pilyavskyy, “Ising model and the positive orthogonal Grassmannian”, Duke Math. J., 169:10 (2020), 1877–1942 | DOI | MR

[3] T. Lam, P. Pilyavskyy, “Electrical networks and Lie theory”, Algebra Number Theory, 9:6 (2015), 1401–1418 | DOI | MR

[4] T. Lam, “Electroid varieties and a compactification of the space of electrical networks”, Adv. Math., 338 (2018), 549–600 | DOI | MR

[5] S. Chepuri, T. George, D. E. Speyer, Electrical networks and Lagrangian Grassmannians, arXiv: 2106.15418

[6] B. Bychkov, V. Gorbounov, A. Kazakov, D. Talalaev, Electrical networks, Lagrangian Grassmannians and symplectic groups, arXiv: 2109.13952

[7] A. B. Zamolodchikov, “Uravneniya tetraedrov i integriruemye sistemy v trekhmernom prostranstve”, ZhETF, 79:2 (1980), 641–664 | MR

[8] S. Sergeev, chastnoe soobschenie, 2020

[9] V. Gorbounov, D. Talalaev, “Electrical varieties as vertex integrable statistical models”, J. Phys. A: Math. Theor., 53:45 (2020), 454001, 28 pp. | DOI | MR

[10] E. B. Curtis, D. Ingerman, J. A. Morrow, “Circular planar graphs and resistor networks”, Linear Algebra Appl., 283:1–3 (1998), 115–150 | DOI | MR

[11] E. B. Curtis, J. A. Morrow, Inverse Problems for Electrical Networks, Series on Applied Mathematics, 13, World Sci., Singapore, 2000 | DOI

[12] A. Berenstein, A. Gainutdinov, V. Gorbounov, chastnoe soobschenie, 2021