Geodesic
Integrals of tau functions: A round dance tau function and multimatrix integrals
Teoretičeskaâ i matematičeskaâ fizika, Tome 215 (2023) no. 3, pp. 377-387
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A multimatrix model can be built on two ingredients: the choice of an embedded graph and the choice of the integrand, that is, a tau function. We compare the simplest nontrivial tau function of the Toda lattice and the simplest nontrivial tau function of the $\mathcal N$-component Toda lattice in the context of their application to multimatrix integrals.
Keywords: tau function, special solution of the multicomponent KP hierarchy, embedded graph, Hurwitz number. 
@article{TMF_2023_215_3_a2,
     author = {A. Yu. Orlov},
     title = {Integrals of tau functions: {A~round} dance tau function and multimatrix integrals},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {377--387},
     year = {2023},
     volume = {215},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2023_215_3_a2/}
}
TY  - JOUR
AU  - A. Yu. Orlov
TI  - Integrals of tau functions: A round dance tau function and multimatrix integrals
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2023
SP  - 377
EP  - 387
VL  - 215
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2023_215_3_a2/
LA  - ru
ID  - TMF_2023_215_3_a2
ER  - 
%0 Journal Article
%A A. Yu. Orlov
%T Integrals of tau functions: A round dance tau function and multimatrix integrals
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2023
%P 377-387
%V 215
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2023_215_3_a2/
%G ru
%F TMF_2023_215_3_a2
A. Yu. Orlov. Integrals of tau functions: A round dance tau function and multimatrix integrals. Teoretičeskaâ i matematičeskaâ fizika, Tome 215 (2023) no. 3, pp. 377-387. http://geodesic.mathdoc.fr/item/TMF_2023_215_3_a2/

[1] A. K. Pogrebkov, V. N. Sushko, “Kvantovanie $(\sin\varphi)_2$-vzaimodeistviya v terminakh fermionnykh peremennykh”, TMF, 24:3 (1975), 425–429 | DOI

[2] M. Sato, Y. Sato, “On Hirota's bilinear equations. II”, Res. Inst. Math. Sci. Kôkyûroku, 414 (1981), 181–202 (in Japanese)

[3] M. Jimbo, T. Miwa, “Solitons and infinite dimensional Lie algebras”, Publ. Res. Inst. Math. Sci., 19:3 (1983), 943–1001 | DOI | MR | Zbl

[4] A. Yu. Orlov, D. M. Scherbin, Fermionic representation for basic hypergeometric functions related to Schur polynomials, arXiv: nlin/0001001

[5] J. Harnad, A. Yu. Orlov, “Matrix integrals as Borel sums of Schur function expansions”, SPT 2002: Symmetries and Perturbation Theory (Cala Gonone, Sardinia, Italy, May 19–26, 2002), eds. S. Abenda, G. Gaeta, S. Walcher, World Sci., River Edge, NJ, 2002, 116–123 | DOI | MR

[6] A. Yu. Orlov, “Deformed Ginibre ensembles and integrable systems”, Phys. Lett. A, 378:4 (2014), 319–328 | DOI | MR

[7] A. Yu. Orlov, “Soliton theory, symmetric functions and matrix integrals”, Acta Appl. Math., 86:1–2 (2005), 131–158, arXiv: nlin/0207030 | DOI | MR | Zbl

[8] A. Yu. Orlov, “New solvable matrix integrals”, Internat. J. Modern Phys. A, 19:supp02 (2004), 276–293 | DOI | MR | Zbl

[9] A. Yu. Orlov, “Chisla Gurvitsa i proizvedeniya sluchainykh matrits”, TMF, 192:3 (2017), 395–443 | DOI | MR

[10] A. Yu. Orlov, “$2D$ Yang–Mills theory and tau functions”, Geometric Methods in Physics XXXVIII, Trends in Mathematics, eds. P. Kielanowski, A. Odzijewicz, E. Previato, Birkhäuser, Cham, 2020, 221–250 | DOI | MR

[11] E. Witten, “On quantum gauge theories in two dimensions”, Comun. Math. Phys., 141:1 (1991), 153–209 | DOI | MR

[12] N. Amburg, A. Orlov, D. Vasiliev, “On products of random matrices”, Entropy, 22:9 (2020), 972, 36 pp. | DOI | MR

[13] A. Yu. Orlov, T. Shiota, K. Takasaki, Pfaffian structures and certain solutions to BKP hierarchies I. Sums over partitions, arXiv: 1201.4518

[14] S. Kharchev, A. Marshakov, A. Mironov, A. Morozov, “Generalized Kazakov–Migdal–Kontsevich model: group theory aspects”, Internat. J. Modern Phys. A, 10:14 (1995), 2015–2051 | DOI | MR

[15] S. Kharchev, Kadomtsev–Petviashvili hierarchy and generalized Kontsevich model, arXiv: hep-th/9810091

[16] Dzh. Kharnad, A. Yu. Orlov, “Skalyarnye proizvedeniya simmetricheskikh funktsii i matrichnye integraly”, TMF, 137:3 (2003), 375–392 | DOI | DOI | MR | Zbl

[17] A. Yu. Orlov, T. Shiota, “Schur function expansion for normal matrix model and associated discrete matrix models”, Phys. Lett. A, 343:5 (2005), 384–396 | DOI | MR | Zbl

[18] A. Yu. Orlov, T. Shiota, K. Takasaki, “Pfaffian structures and certain solutions to BKP hierarchies II. Multiple integrals”, arXiv: 1611.02244

[19] S. M. Natanzon, A. Yu. Orlov, “BKP and projective Hurwitz numbers”, Lett. Math. Phys., 107:6 (2017), 1065–1109, arXiv: 1501.01283 | DOI | MR

[20] V. Kac, J. van de Leur, “The geometry of spinors and the multicomponent BKP and DKP hierarchies”, The Bispectral Problem (Montréal, Canada, March 1997), CRM Proceedings and Lecture Notes, 14, eds. J. Harnad, A. Kasman, AMS, Providence, RI, 1998, 159–202 | DOI | MR | Zbl

[21] V. A. Kazakov, M. Staudacher, T. Wynter, “Character expansion methods for matrix models of dually weighted graphs”, Commun. Math. Phys., 177:2 (1996), 451–468, arXiv: hep-th/9502132 | DOI | MR | Zbl

[22] J. Harnad, A. Yu. Orlov, “Fermionic construction of partition functions for two matrix models and perturbative Schur functions expansions”, J. Phys. A.: Math. Gen., 39:28 (2006), 8783–8809 | DOI | MR | Zbl

[23] Dzh. Kharnad, I. V. van de Ler, A. Yu. Orlov, “Kratnye summy i integraly kak tau-\allowbreakfunktsii neitralnoi ierarkhii Kadomtseva–Petviashvili”, TMF, 168:1 (2011), 112–124 | DOI | DOI | MR

[24] A. Yu. Orlov, New solvable matrix models III, arXiv: 2112.14741

[25] A. Alexandrov, “Matrix models for random partitions”, Nucl. Phys. B, 851:3 (2011), 620–650 | DOI | MR | Zbl

[26] K. Ueno, K. Takasaki, “Toda lattice hierarchy”, Group Representations and Systems of Differential Equations (University of Tokyo, Japan, December 20–27, 1982), Advanced Studies in Pure Mathematics, 4, ed. K. Okamoto, North-Holland, Amsterdam, 1984, 1–95 | DOI | MR | Zbl

[27] A. D. Mironov, A. Yu. Morozov, S. M. Natanzon, “Polnyi nabor operatorov razrezaniya i skleiki v teorii Gurvitsa–Kontsevicha”, TMF, 166:1 (2011), 3–27, arXiv: 0904.4227 | DOI | MR

[28] A. Alexandrov, A. Mironov, A. Morozov, S. Natanzon, “Integrability of Hurwitz partition functions”, J. Phys. A: Math. Theor., 45:4 (2012), 045209, 10 pp., arXiv: 1103.4100 | DOI | MR

[29] I. Makdonald, Simmetricheskie funktsii i mnogochleny Kholla, Mir, M., 1984 | MR

[30] A. V. Mikhailov, “Ob integriruemosti dvumernogo obobscheniya tsepochki Toda”, Pisma v ZhETF, 30:7 (1979), 443–448; A. V. Mikhailov, M. A. Olshanetsky, A. M. Perelomov, “Two-dimensional generalized Toda lattice”, Commun. Math. Phys., 79:4 (1981), 473–488 | DOI | MR

[31] K. Takasaki, “Initial value problem for the Toda lattice hierarchy”, Group Representations and Systems of Differential Equations (University of Tokyo, December 20–27, 1982), Advanced Studies in Pure Mathematics, 4, ed. K. Okamoto, North-Holland, Amsterdam, 1984, 139–163 | DOI | MR | Zbl

[32] T. Takebe, “Representation theoretical meaning of initial value problem for the Toda lattice hierarchy. I”, Lett. Math. Phys., 21:1 (1991), 77–84 | DOI | MR

[33] A. Mironov, A. Morozov, “Hook variables: cut-and-join operators and $\tau$ functions”, Phys. Lett. B, 804 (2020), 135362, 13 pp., arXiv: 1912.00635 | DOI | MR

[34] M. L. Meta, Sluchainye matritsy, MTsNMO, M., 2012 | MR

[35] E. Strahov, “Differential equations for singular values of products of Ginibre random matrices”, J. Phys. A: Math. Theor., 47:32 (2014), 325203, 27 pp., arXiv: 1403.6368 | DOI | MR

[36] J. R. Ipsen, Products of independent Gaussian random matrices, arXiv: 1510.06128

[37] S. M. Natanzon, A. Yu. Orlov, “Hurwitz numbers from matrix integrals over Gaussian measure”, Integrability, Quantization, and Geometry: I. Integrable Systems, Proceedings of Symposia in Pure Mathematics, 103, eds. S. Novikov, I. Krichever, O. Ogievetsky, S. Shlosman, AMS, Providence, RI, 2021, 337–377, arXiv: 2002.00466 | DOI | MR

[38] S. M. Natanzon, A. Yu. Orlov, “Chisla Gurvitsa, poluchayuschiesya iz feinmanovskikh diagramm”, TMF, 204:3 (2020), 396–429, arXiv: 2006.07396 | DOI | DOI

[39] S. K. Lando, A. K. Zvonkin, Grafy na poverkhnostyakh i ikh prilozheniya, MTsNMO, M., 2010 | DOI | MR

[40] A. D. Mironov, A. Yu. Morozov, S. M. Natanzon, “Integrability properties of Hurwitz partition functions. II. Multiplication of cut-and-join operators and WDVV equations”, JHEP, 11 (2011), 097, 32 pp., arXiv: 1108.0885 | DOI | MR

[41] A. D. Mironov, A. Yu. Morozov, S. M. Natanzon, “Algebra of differential operators associated with Young diagramms”, J. Geom. Phys., 62:2 (2012), 148–155, arXiv: 1012.0433 | DOI | MR

[42] A. Alexandrov, A. Mironov, A. Morozov, S. Natanzon, “On KP-integrable Hurwitz functions”, JHEP, 11 (2014), 080, 30 pp., arXiv: 1405.1395 | DOI | MR

[43] A. Yu. Orlov, D. M. Scherbin, “Gipergeometricheskie resheniya solitonnykh uravnenii”, TMF, 128:1 (2001), 84–108, arXiv: nlin/0001001 | DOI | DOI | MR | Zbl

[44] Ya. Amborn, L. O. Chekhov, “Matrichnaya model dlya gipergeometricheskikh chisel Gurvitsa”, TMF, 181:3 (2014), 421–43, arXiv: 1409.3553 | DOI | DOI | MR

[45] I. P. Goulden, D. M. Jackson, “The KP hierarchy, branched covers and triangulations”, Adv. Math., 219:3 (2008), 932–951 | DOI | MR

[46] M. E. Kazaryan, S. K. Lando, “Kombinatornye resheniya integriruemykh ierarkhii”, UMN, 70:3(423) (2015), 77–106, arXiv: 1512.07172 | DOI | DOI | MR | Zbl

[47] M. Guay-Paquet, J. Harnad, “2D Toda $\tau$-functions as combinatorial generating functions”, Lett. Math. Phys., 105:6 (2015), 827–852 | DOI | MR | Zbl

[48] J. Harnad, A. Yu. Orlov, “Hypergeometric $\tau$-functions, Hurwitz numbers and enumeration of paths”, Commun. Math. Phys., 338:1 (2015), 267–284, arXiv: 1407.7800 | DOI | MR

[49] S. M. Natanzon, A. Yu. Orlov, “BKP and projective Hurwitz numbers”, Lett. Math. Phys., 107:6 (2017), 1065–1109, arXiv: 1501.01283 | DOI | MR