arXiv subjects

Anton Dzhamay

Publications and source records attributed to Anton Dzhamay.

At least 19 recordsLinked to original sources

arXiv2026

On the discrete Painlevé equivalence problem, non-conjugate translations and nodal curves

We consider several examples of nonautonomous systems of difference equations coming from semi-classical orthogonal polynomials via recurrence coefficients and ladder operators, with respect to various generalisations of Laguerre and Meixner weights. We identify these as discrete Painlevé equations and establish their types in the Sakai classification scheme in terms of the associated rational surfaces. In particular, we find examples which come from different weights and share a common surface type $D_5^{(1)}$ but are inequivalent in two ways. First, their dynamics are generated by non-conjugate elements of $\widehat{W}(A_3^{(1)})$. Second, some of the examples have associated surfaces being non-generic in the sense of having nodal curves. The symmetries of these examples form subgroups of the generic symmetry group, which we compute. In particular, we find $(W(A_1^{(1)})\times W(A_1^{(1)}))\rtimes \mathbb{Z}/2\mathbb{Z}$. These examples give further weight to the argument that any correspondence between different weights and the Sakai classification should make use of the refined version of the discrete Painlevé equivalence problem, which takes into account not just surface type, but also the group elements generating the dynamics as well as parameter constraints, e.g. those corresponding to nodal curves.

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arXiv2025

Orthogonal Polynomials for the Gaussian Weight with a Jump and Discrete Painlevé Equations

We describe a refined version of the discrete Painlevé identification problem that emphasizes the importance on going beyond just the surface type in describing a discrete Painlevé dynamic. We give an example of solving such identification problem for a particular recurrence relation that previously appeared in studying orthogonal polynomials for the Gaussian weight with a single jump.

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arXiv2025

On some examples of identifying Painlevé equations using the geometry of the Okamoto space of initial conditions

In this short note we give two examples of using the algebro-geometric theory of Painlevé equations to solve the Painlevé identification problem. The equations that we consider were recently obtained by M. van der Put and J. Top in their study of a certain ansatz of isomonodromic deformations of linear ODEs. We provide explicit coordinate transformations identifying these examples with standard form of some Painlevé equations and also explicitly identify their Hamiltonians.

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arXiv2025

Special solutions of a discrete Painlevé equation for quantum minimal surfaces

We consider solutions of a discrete Painlevé equation arising from a construction of quantum minimal surfaces by Arnlind, Hoppe and Kontsevich, and in earlier work of Cornalba and Taylor on static membranes. While the discrete equation admits a continuum limit to the continuous Painlevé I equation, we find that it has the same space of initial values as the Painlevé V equation with certain specific parameter values. We further explicitly show how each iteration of this discrete Painlevé I equation corresponds to a certain composition of Bäcklund transformations for Painlevé V, as was first remarked in work by Tokihiro, Grammaticos and Ramani. In addition, we show that some explicit special function solutions of Painlevé V, written in terms of modified Bessel functions, yield the unique positive solution of the initial value problem required for quantum minimal surfaces.

math-ph↗

arXiv2025

Recurrence relations for the generalized Laguerre and Charlier orthogonal polynomials and discrete Painlevé equations on the $D_{6}^{(1)}$ Sakai surface

This paper concerns the discrete version of the Painlevé identification problem, i.e., how to recognize a certain recurrence relation as a discrete Painlevé equation. Often some clues can be seen from the setting of the problem, e.g., when the recurrence is connected with some differential Painlevé equation, or from the geometry of the configuration of indeterminate points of the equation. The main message of our paper is that, in fact, this only allows us to identify the configuration space of the dynamic system, but not the dynamics themselves. The refined version of the identification problem lies in determining, up to the conjugation, the translation direction of the dynamics, which in turn requires the full power of the geometric theory of Painlevé equations. To illustrate this point, in this paper we consider two examples of such recurrences that appear in the theory of orthogonal polynomials. We choose these examples because they get regularized on the same family of Sakai surfaces, but at the same time are not equivalent, since they result in non-equivalent translation directions. In addition, we show the effectiveness of a recently proposed identification procedure for discrete Painlevé equations using Sakai's geometric approach for answering such questions. In particular, this approach requires no a priori knowledge of a possible type of the equation.

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arXiv2025

What is the symmetry group of a d-${\text{P}_{\mathrm{II}}}$ discrete Painlevé equation?

The symmetry group of a (discrete) Painlevé equation provides crucial information on the properties of the equation. In this paper we argue against the commonly-held belief that the symmetry group of a given equation is solely determined by its surface type as given in the famous Sakai classification. We will dispel this misconception on a specific example of a d-${\text{P}_{\mathrm{II}}}$ equation which corresponds to a half-translation on the root lattice dual to its surface-type root lattice, but which becomes a genuine translation on a sub-lattice thereof that corresponds to its real symmetry group. The latter fact is shown in two different ways: first by a brute force calculation and second through the use of normalizer theory, which we believe to be an extremely useful tool for this purpose. We finish the paper with the analysis of a sub-case of our main example which arises in the study of gap probabilities for Freud unitary ensembles, and the symmetry group of which is even further restricted due to the appearance of a nodal curve on the surface on which the equation is regularized.

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arXiv2024

On the recurrence coefficients for the $q$-Laguerre weight and discrete Painlevé equations

We study the dependence of recurrence coefficients in the three-term recurrence relation for orthogonal polynomials with a certain deformation of the $q$-Laguerre weight on the degree parameter $n$. We show that this dependence is described by a discrete Painlevé equation on the family of $A_{5}^{(1)}$ Sakai surfaces, but this equation is different from the standard examples of discrete Painlevé equations of this type and instead is a composition of two such. This case study is a good illustration of the effectiveness of a recently proposed geometric identification scheme for discrete Painlevé equations.

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arXiv2023

Different Hamiltonians for differential Painlevé equations and their identification using a geometric approach

It is well-known that differential Painlevé equations can be written in a Hamiltonian form. However, a coordinate form of such representation is far from unique -- there are many very different Hamiltonians that result in the same differential Painlevé equation. Recognizing a Painlevé equation, for example when it appears in some applied problem, is known as the \emph{Painlevé equivalence problem}, and the question that we consider here is the Hamiltonian form of this problem. Making such identification explicit, on the level of coordinate transformations, can be very helpful for an applied problem, since it gives access to the wealth of known results about Painlevé equations, such as the structure of the symmetry group, special solutions for special values of the parameters, and so on. It can also provide an explicit link between different problems that have the same underlying structure. In this paper we describe a systematic procedure for finding changes of coordinates that trasform different Hamiltonian representations of a Painlevé equation into some chosen canonical form. Our approach is based on Sakai's geometric theory of Painlevé equations. We explain this procedure in detail for the fourth differential ${\text{P}_{\mathrm{IV}}}$ equation, and also give a brief summary of some known examples for ${\text{P}_{\mathrm{V}}}$ and ${\text{P}_{\mathrm{VI}}}$ cases. It is clear that this approach can easily be adapted to other examples as well, so we expect our paper to be a useful reference for some of the realizations of Okamoto spaces of initial conditions for Painlevé equations.

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arXiv2021

Differential equations for the recurrence coefficients of semi-classical orthogonal polynomials and their relation to the Painlevé equations via the geometric approach

In this paper we present a general scheme for how to relate differential equations for the recurrence coefficients of semi-classical orthogonal polynomials to the Painlevé equations using the geometric framework of the Okamoto Space of Initial Conditions. We demonstrate this procedure in two examples. For semi-classical Laguerre polynomials appearing in \cite{HC17}, we show how the recurrence coefficients are connected to the fourth Painlevé equation. For discrete orthogonal polynomials associated with the hypergeometric weight appearing in \cite{FVA18} we discuss the relation of the recurrence coefficients to the sixth Painlevé equation, extending the results of \cite{DFS19}, where a similar approach was used for a discrete system for the same recurrence coefficients. Though the discrete and differential systems here share the same geometry, the construction of the space of initial conditions from the differential system is different and reveals extra considerations that must be made. We also discuss a number of related topics in the context of the geometric approach, such as Hamiltonian forms of the differential equations for the recurrence coefficients, Riccati solutions for special parameter values, and associated discrete Painlevé equations.

math.CA↗

arXiv2020

Gap Probabilities in the Laguerre Unitary Ensemble and Discrete Painlevé Equations

In this paper we study a certain recurrence relation, that can be used to generate ladder operators for the Laguerre Unitary ensemble, from the point of view of Sakai's geometric theory of Painlevé equations. On one hand, this gives us one more detailed example of the appearance of discrete Painlevé equations in the theory of orthogonal polynomials. On the other hand, it serves as a good illustration of the effectiveness of a recently proposed procedure on how to reduce such recurrences to some canonical discrete Painlevé equations.

nlin.SI↗

arXiv2019

Recurrence coefficients for discrete orthogonal polynomials with hypergeometric weight and discrete Painlevé equations

Over the last decade it has become clear that discrete Painlevé equations appear in a wide range of important mathematical and physical problems. Thus, the question of recognizing a given non-autonomous recurrence as a discrete Painlevé equation and determining its type according to Sakai's classification scheme, understanding whether it is equivalent to some known (model) example, and especially finding an explicit change of coordinates transforming it to such an example, becomes one of the central ones. Fortunately, Sakai's geometric theory provides an almost algorithmic procedure for answering this question. In this paper we illustrate this procedure by studying an example coming from the theory of discrete orthogonal polynomials. There are many connections between orthogonal polynomials and Painlevé equations, both differential and discrete. In particular, often the coefficients of three-term recurrence relations for discrete orthogonal polynomials can be expressed in terms of solutions of discrete Painlevé equations. In this work we study discrete orthogonal polynomials with general hypergeometric weight and show that their recurrence coefficients satisfy, after some change of variables, the standard discrete Painlevé-V equation. We also provide an explicit change of variables transforming this equation to the standard form.

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arXiv2019

$q$-Racah ensemble and $q$-P$\left(E_7^{(1)}/A_{1}^{(1)}\right)$ discrete Painlevé equation

The goal of this paper is to investigate the missing part of the story about the relationship between the orthogonal polynomial ensembles and Painlevé equations. Namely, we consider the $q$-Racah polynomial ensemble and show that the one-interval gap probabilities in this case can be expressed through a solution of the discrete $q$-P$\left(E_7^{(1)}/A_{1}^{(1)}\right)$ equation. Our approach also gives a new Lax pair for this equation. This Lax pair has an interesting additional involutive symmetry structure.

math-ph↗

arXiv2018

On Some Applications of Sakai's Geometric Theory of Discrete Painlevé Equations

Although the theory of discrete Painlevé (dP) equations is rather young, more and more examples of such equations appear in interesting and important applications. Thus, it is essential to be able to recognize these equations, to be able to identify their type, and to see where they belong in the classification scheme. The definite classification scheme for dP equations was proposed by H. Sakai, who used geometric ideas to identify 22 different classes of these equations. However, in a major contrast with the theory of ordinary differential Painlevé equations, there are infinitely many non-equivalent discrete equations in each class. Thus, there is no general form for a dP equation in each class, although some nice canonical examples in each equation class are known. The main objective of this paper is to illustrate that, in addition to providing the classification scheme, the geometric ideas of Sakai give us a powerful tool to study dP equations. We consider a very complicated example of a dP equation that describes a simple Schlesinger transformation of a Fuchsian system and we show how this equation can be identified with a much simpler canonical example of the dP equation of the same type and moreover, we give an explicit change of coordinates transforming one equation into the other. Among our main tools are the birational representation of the affine Weyl symmetry group of the equation and the period map. Even though we focus on a concrete example, the techniques that we use are general and can be easily adapted to other examples.

math-ph↗

arXiv2017

Fiber-dependent deautonomization of integrable 2D mappings and discrete Painlevé equations

It is well known that two-dimensional mappings preserving a rational elliptic fibration, like the Quispel-Roberts-Thompson mappings, can be deautonomized to discrete Painlevé equations. However, the dependence of this procedure on the choice of a particular elliptic fiber has not been sufficiently investigated. In this paper we establish a way of performing the deautonomization for a pair of an autonomous mapping and a fiber. %By choosing a particular Starting from a single autonomous mapping but varying the type of a chosen fiber, we obtain different types of discrete Painlevé equations using this deautonomization procedure. We also introduce a technique for reconstructing a mapping from the knowledge of its induced action on the Picard group and some additional geometric data. This technique allows us to obtain factorized expressions of discrete Painlevé equations, including the elliptic case. Further, by imposing certain restrictions on such non-autonomous mappings we obtain new and simple elliptic difference Painlevé equations, including examples whose symmetry groups do not appear explicitly in Sakai's classification.

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arXiv2014

Geometric Analysis of Reductions from Schlesinger Transformations to Difference Painlevé Equations

We present two examples of reductions from the evolution equations describing discrete Schlesinger transformations of Fuchsian systems to difference Painlevé equations: difference Painlevé equation d-$P\left({A}_{2}^{(1)*}\right)$ with the symmetry group ${E}^{(1)}_{6}$ and difference Painlevé equation d-$P\left({A}_{1}^{(1)*}\right)$ with the symmetry group ${E}^{(1)}_{7}$. In both cases we describe in detail how to compute their Okamoto space of the initial conditions and emphasize the role played by geometry in helping us to understand the structure of the reduction, a choice of a good coordinate system describing the equation, and how to compare it with other instances of equations of the same type.

math-ph↗

arXiv2014

Discrete Schlesinger Transformations, their Hamiltonian Formulation, and Difference Painlevé Equations

Schlesinger transformations are algebraic transformations of a Fuchsian system that preserve its monodromy representation and act on the characteristic indices of the system by integral shifts. One of the important reasons to study such transformations is the relationship between Schlesinger transformations and discrete Painlevé equations; this is also the main theme behind our work. We derive \emph{discrete Schlesinger evolution equations} describing discrete dynamical systems generated by elementary Schlesinger transformations and give their discrete Hamiltonian description w.r.t.~the standard symplectic structure on the space of Fuchsian systems. As an application, we compute explicitly two examples of reduction from Schlesinger transformations to difference Painlevé equations. The first example, d-$P\big(D_{4}^{(1)}\big)$ (or difference Painlevé V), corresponds to Bäcklund transformations for continuous $P_{\text{VI}}$. The second example, d-$P\big(A_{2}^{(1)*}\big)$ (with the symmetry group $E_{6}^{(1)}$), is purely discrete. We also describe the role played by the geometry of the Okamoto space of initial conditions in comparing different equations of the same type.

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arXiv2013

Combinatorics of Matrix Factorizations and Integrable Systems

We study relations between the eigenvectors of rational matrix functions on the Riemann sphere. Our main result is that for a subclass of functions that are products of two elementary blocks it is possible to represent these relations in a combinatorial-geometric way using a diagram of a cube. In this representation, vertices of the cube represent eigenvectors, edges are labeled by differences of locations of zeroes and poles of the determinant of our matrix function, and each face corresponds to a particular choice of a coordinate system on the space of such functions. Moreover, for each face this labeling encodes, in a neat and efficient way, a generating function for the expressions of the remaining four eigenvectors that label the opposing face of the cube in terms of the coordinates represented by the chosen face. The main motivation behind this work is that when our matrix is a Lax matrix of a discrete integrable system, such generating functions can be interpreted as Lagrangians of the system, and a choice of a particular face corresponds to a choice of the direction of the motion.

nlin.SI↗

arXiv2013

Factorizations of Rational Matrix Functions with Application to Discrete Isomonodromic Transformations and Difference Painlevé Equations

We study factorizations of rational matrix functions with simple poles on the Riemann sphere. For the quadratic case (two poles) we show, using multiplicative representations of such matrix functions, that a good coordinate system on this space is given by a mix of residue eigenvectors of the matrix and its inverse. Our approach is motivated by the theory of discrete isomonodromic transformations and their relationship with difference Painlevé equations. In particular, in these coordinates, basic isomonodromic transformations take the form of the discrete Euler-Lagrange equations. Secondly we show that dPV equations, previously obtained in this context by D. Arinkin and A. Borodin, can be understood as simple relationships between the residues of such matrices and their inverses.

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