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## Комбинаторные решения интегрируемых иерархий

Russian Mathematical Surveys. 2015. Т. 70. № 3. С. 70-106.

This paper reviews modern approaches to the construction of formal solutions to integrable hierarchies of mathematical physics whose coefficients are answers to various enumerative problems. The relationship between these approaches and the combinatorics of symmetric groups and their representations is explained. Applications of the results to the construction of efficient computations in problems related to models of quantum field theories are described.

Bychkov B., Dunin-Barkowski P., Kazaryan M. et al., Topological recursion for Kadomtsev-Petviashvili tau functions of hypergeometric type / Cornell University. Series math "arxiv.org". 2020. No. 2012.14723.

We study the n-point differentials corresponding to Kadomtsev-Petviashvili tau functions of hypergeometric type (also known as Orlov-Scherbin partition functions), with an emphasis on their ℏ2-deformations and expansions.
Under the naturally required analytic assumptions, we prove certain higher loop equations that, in particular, contain the standard linear and quadratic loop equations, and thus imply the blobbed topological recursion. We ...

Added: April 20, 2022

Boris Bychkov, Petr Dunin-Barkowski, Maxim Kazarian et al., Journal de l'Ecole Polytechnique - Mathematiques 2022 Vol. 9 P. 1121-1158

We derive a new explicit formula in terms of sums over graphs for the n-point correlation functions of general formal weighted double Hurwitz numbers coming from the Kadomtsev–Petviashvili tau functions of hypergeometric type (also known as Orlov–Scherbin partition functions). Notably, we use the change of variables suggested by the associated spectral curve, and our formula ...

Added: August 6, 2022

Dunin-Barkowski P., Kramer R., Popolitov A. et al., Journal of Geometry and Physics 2019 Vol. 137 P. 1-6

We give a new proof of the cut-and-join equation for the monotone Hurwitz numbers, derived first by Goulden, Guay-Paquet, and Novak. The main interest in this particular equation is its close relation to the quadratic loop equation in the theory of spectral curve topological recursion, and we recall this motivation giving a new proof of ...

Added: February 20, 2019

Kharchev S., Levin A., Olshanetsky M. et al., Journal of Mathematical Physics 2018 Vol. 59 No. 103509 P. 1-36

We define the quasi-compact Higgs G -bundles over singular curves introduced in our previous paper for the Lie group SL(N). The quasi-compact structure means that the automorphism groups of the bundles are reduced to the maximal compact subgroups of G at marked points of the curves. We demonstrate that in particular cases, this construction leads ...

Added: October 20, 2018

Marshakov A., Fock V., Loop groups, Clusters, Dimers and Integrable systems / Cornell University. Series math "arxiv.org". 2014.

We describe a class of integrable systems on Poisson submanifolds of the affine Poisson-Lie groups PGLˆ(N), which can be enumerated by cyclically irreducible elements the co-extended affine Weyl groups (Wˆ×Wˆ)♯. Their phase spaces admit cluster coordinates, whereas the integrals of motion are cluster functions. We show, that this class of integrable systems coincides with the ...

Added: October 29, 2014

Zabrodin A., Journal of Physics A: Mathematical and Theoretical 2013 Vol. 46 No. 18 P. 185203

We study the integrable structure of the 2D Laplacian growth problem with zero surface tension in an infinite channel with periodic boundary conditions in a transverse direction. Similarly to the Laplacian growth in radial geometry, this problem can be embedded into the 2D Toda lattice hierarchy in the zero dispersion limit. However, the relevant solution ...

Added: April 29, 2013

Shapiro B., Yurii Burman, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 2019 Vol. XIX No. 1 P. 155-167

For a point p of the complex projective plane and a triple (g,d,l) of non-negative integers we define a Hurwitz--Severi number H(g,d,l) as the number of generic irreducible plane curves of genus g and degree d+l having an l-fold node at p and at most ordinary nodes as singularities at the other points, such that the ...

Added: April 14, 2017

Buryak A., Dubrovin B., Guere J. et al., International Mathematics Research Notices 2020 Vol. 2020 No. 24 P. 10381-10446

In this paper we study various aspects of the double ramification (DR) hierarchy, introduced by the 1st author, and its quantization. We extend the notion of tau-symmetry to quantum integrable hierarchies and prove that the quantum DR hierarchy enjoys this property. We determine explicitly the genus 1 quantum correction and, as an application, compute completely the quantization ...

Added: April 21, 2020

Derbyshev A. E., Povolotsky A. M., Priezzhev V. B., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2015 Vol. 91 P. 022125

The generalized totally asymmetric exclusion process (TASEP) [J. Stat. Mech. (2012) P05014] is an integrable generalization of the TASEP equipped with an interaction, which enhances the clustering of particles. The process interpolates between two extremal cases: the TASEP with parallel update and the process with all particles irreversibly merging into a single cluster moving as ...

Added: February 19, 2015

Krichever I. M., Функциональный анализ и его приложения 2012 Т. 46 № 2 С. 37-51

Using meromorphic differentials with real periods, we prove Arbarello's conjecture that any compact complex cycle of dimension g−n in the moduli space M_g of smooth algebraic curves of genus g must intersect the locus of curves having a Weierstrass point of order at most n. ...

Added: April 17, 2014

Burman Y. M., Zvonkine D., European Journal of Combinatorics 2010 Vol. 31 No. 1 P. 129-144

Consider factorizations into transpositions of an n-cycle in the symmetric group Sn. To every such factorization we assign a monomial in variables wij that retains the transpositions used, but forgets their order. Summing over all possible factorizations of n-cycles we obtain a polynomial that happens to admit a closed expression. From this expression we deduce ...

Added: November 7, 2012

Dunin-Barkowski P., Lewanski D., Popolitov A. et al., Journal of London Mathematical Society 2015 Vol. 92 No. 3 P. 547-565

In this paper, we present an example of a derivation of an ELSV-type formula using the methods of topological recursion. Namely, for orbifold Hurwitz numbers we give a new proof of the spectral curve topological recursion, in the sense of Chekhov, Eynard and Orantin, where the main new step compared to the existing proofs is ...

Added: November 16, 2015

Bychkov B., Dunin-Barkowski P., Shadrin S., European Journal of Combinatorics 2020 Vol. 90 P. 103184

In this paper we prove, in a purely combinatorial-algebraic way, a structural quasi-polynomiality property for the Bousquet-Mélou–Schaeffer numbers. Conjecturally, this property should follow from the Chekhov–Eynard–Orantin topological recursion for these numbers (or, to be more precise, the Bouchard–Eynard version of the topological recursion for higher order critical points), which we derive in this paper from ...

Added: September 22, 2020

Yurii Burman, Shapiro B., On Hurwitz--Severi numbers / Cornell University. Series math "arxiv.org". 2016. No. 06935.

For a point p in a complex projective plane and a triple (g,d,l) of non-negative
integers we define a plane Hurwitz number of the Severi variety
W_{g,d,l} consisting of all reduced irreducible plane curves of
genus g and degree d+l having an l-fold node at p and at
most ordinary nodes as singularities at the other points. In the ...

Added: July 5, 2016

Dunin-Barkowski Petr, Kazarian Maxim, Orantin N. et al., Advances in Mathematics 2015 Vol. 279 P. 67-103

In this paper we give a new proof of the ELSV formula. First, we refine an argument of Okounkov and Pandharipande in order to prove (quasi-)polynomiality of Hurwitz numbers without using the ELSV formula (the only way to do that before used the ELSV formula). Then, using this polynomiality we give a new proof of ...

Added: September 24, 2015

Costa A., Sergey Natanzon, Shapiro B., Annales Academiae Scientiarum Fennicae Mathematica 2018 Vol. 43 P. 349-363

In this article, to each generic real meromorphic function (i.e., having only simple branch points in the appropriate sense) we associate a certain combinatorial gadget which we call the park of a function. We show that the park determines the topological type of the generic real meromorphic function and the set of parks produce an stratification ...

Added: March 4, 2018

Povolotsky A. M., Journal of Statistical Mechanics: Theory and Experiment 2019 No. 074003 P. 1-22

We establish the exact laws of large numbers for two time additive quantities in the raise and peel model, the number of tiles removed by avalanches and the number of global avalanches happened by given time. The validity of conjectures for the related stationary state correlation functions then follow. The proof is based on the ...

Added: October 8, 2019

Khoroshkin S. M., Tsuboi Z., Journal of Physics A: Mathematical and Theoretical 2014 Vol. 47 P. 1-11

We consider the 'universal monodromy operators' for the Baxter Q-operators. They are given as images of the universal R-matrix in oscillator representation. We find related universal factorization formulas in the Uq(\hat{sl}(2)) case. ...

Added: December 8, 2014

Alexeevski A., Natanzon S. M., American Mathematical Society Translations 2014 Vol. 234 P. 1-12

In 2001 Ivanov and Kerov associated with the infinite permutation group S∞ certain commutative associative algebra A∞ called the algebra of conjugacy classes of partial elements. A standard basis of A∞ islabeled by Yang diagrams of all orders. Mironov, Morozov, Natanzon, 2012, have proved that the completion of A∞ is isomorphic to the direct product ...

Added: April 2, 2014

Marshall I., International Mathematics Research Notices 2015 Vol. 18 P. 8925-8958

A Poisson structure is defined on the space {\mathcal {W}} of twisted polygons in {\mathbb {R}}^{\nu }. Poisson reductions with respect to two Poisson group actions on {\mathcal {W}} are described. The \nu =2 and \nu =3 cases are discussed in detail. Amongst the Poisson structures arising in examples are to be found the lattice ...

Added: November 28, 2014

Mironov A., Morozov A., Natanzon S. M., Journal of Knot Theory and Its Ramifications 2014 Vol. 23 No. 6 P. 1-16

The classical Hurwitz numbers of degree n together with the Hurwitz numbers of the seamed surfaces of degree n give rise to the Klein topological field theory. We extend this construction to the Hurwitz numbers of all degrees at once. The corresponding Cardy-Frobenius algebra is induced by arbitrary Young diagrams and arbitrary bipartite graphs. It ...

Added: April 2, 2014

Nirov Khazret S., Razumov A. V., Journal of Geometry and Physics 2017 Vol. 112 P. 1-28

A detailed construction of the universal integrability objects related to the integrable
systems associated with the quantum loop algebra Uq(L(sl2)) is given. The full proof of the
functional relations in the form independent of the representation of the quantum loop
algebra on the quantum space is presented. The case of the general gradation and general
twisting is treated. The ...

Added: January 29, 2018

Natanzon S. M., Zabrodin A., International Mathematics Research Notices 2015 Vol. 2015 No. 8 P. 2082-2110

We explicitly construct the series expansion for a certain class of solutions to the 2D Toda hierarchy in the zero dispersion limit, which we call symmetric solutions. We express the Taylor coefficients through some universal combinatorial constants and find recurrence relations for them. These results are used to obtain new formulas for the genus 0 ...

Added: April 2, 2014

Natanzon S. M., Orlov A. Y., Theoretical and Mathematical Physics 2020 Vol. 204 No. 3 P. 1166-1194

To obtain a generating function of the most general form for Hurwitz numbers with arbitrary base surfaceand arbitrary ramification profiles, we consider a matrix model constructed according to a graph on anoriented connected surfaceΣwith no boundary. The vertices of this graph, called stars, are small discs,and the graph itself is a clean dessin d’enfants. We ...

Added: September 27, 2020