Discrete integrable systems: difference equations, cluster algebras and probabilistic models

We study the structure of singularities in the discrete Korteweg–deVries equation and its modified sibling. Four different types of singularities are identified. The first type corresponds to localised, ‘confined’, singularities. Two other types of singularities are of infinite extent and consist of oblique lines. The fourth type of singularity corresponds to horizontal strips where the product of the values on vertically adjacent points is equal to 1. Due to its orientation this singularity can, in fact, interact with the other types. This type of singularity was dubbed ‘taishi’. The taishi can interact with singularities of the other two families, giving rise to very rich and quite intricate singularity structures. Nonetheless, these interactions can be described in a compact way through the formulation of a symbolic representation of the dynamics. We give an interpretation of this symbolic representation in terms of a box & ball system related to the ultradiscrete KdV equation.

We introduce two properties that characterise integrable discrete systems: singularity confinement and low growth. The latter is quantified through the dynamical degree, a quantity that is equal to 1 for integrable systems and larger than 1 for non-integrable ones. We show how the structure of singularities conditions the growth properties of a given system. We introduce the full deautonomisation discrete integrability criterion and illustrate its application through concrete examples. Starting from the results of R. Halburd we show how one can obtain the dynamical degree of a given mapping based on its singularity structure. The notion of Diophantine approximation is introduced as a practical way to obtain the dynamical degree. We show how one can obtain the degree growth of a given birational mapping in an algorithmic way using only the information on its singularities. Several examples of second-order mappings are presented and we show how our approach can be extended to higher-orde...

A colored vertex model is a solution of the Yang--Baxter equation based on a higher-rank Lie algebra. These models generalize the famous six-vertex model, which may be viewed in terms of osculating lattice paths, to ensembles of colored paths. By studying certain partition functions within these models, one may define families of multivariate rational functions (or polynomials) with remarkable algebraic features. In these lectures, we will examine a number of these properties:
(a) Exchange relations under the Hecke algebra;
(b) Infinite summation identities of Cauchy-type;
(c) Orthogonality with respect to torus scalar products;
(d) Multiplication rules (combinatorial formulae for structure constants).

Our aim will be to show that all such properties arise very naturally within the algebraic framework provided by the vertex models. If time permits, applications to probability theory will be surveyed.

It is well known that the box-ball system discovered by Takahashi and Satsuma can be obtained by the ultra-discrete analogue of the discrete integrable system, including both the ultra-discrete analogue of the KdV lattice and the ultra-discrete analogue of the Toda lattice. This mini-course will demonstrate that it is possible to derive extended models of the box-ball systems related to the relativistic Toda lattice and the fundamental Toda orbits, which are obtained from the theory of orthogonal polynomials and their extensions. We will first introduce an elementary procedure for deriving box-ball systems from discrete KP equations. Then, we will discuss the relationship between discrete Toda lattices and their extensions based on orthogonal polynomial theory, and outline the exact solutions and ultra-discretization procedures for these systems. Additionally, we will introduce the box-ball system on R, which is obtained by clarifying its relationship with the Pitman transformation in ...

Airy_1 and Airy_2 processes are stationary stochastic processes on the real line that arise in various contexts in integrable probability. In particular, they are obtained as scaling limits of passage time profiles in planar exponential last passage percolation (LPP) models with different initial conditions. In this talk, we shall present law of fractional logarithms with optimal constants for maxima and minima of Airy processes over growing intervals, extending and complementing the work of Pu. We draw upon the recently established sharp tail estimates for various passage times in exponential LPP by Baslingker et al., as well as geometric properties of exponential LPP landscape. The talk is based on a recent work with Riddhipratim Basu
(https://doi.org/10.48550/arXiv.2406.11826).

We will present a method for computing the stationary measures of integrable probabilistic systems on finite domains. Focusing on the example of a well-studied model called last passage percolation, we will describe the stationary measure in various ways, and emphasize the key role played by Schur symmetric functions. The method works as well for other models and their associated families of symmetric functions, suchas Whittaker functions or Hall-Littlewood polynomials. We will also discuss how this is related to the traditional approach for computing stationary measures of interacting particle systems between boundary reservoirs: the matrix product ansatz.

A colored vertex model is a solution of the Yang--Baxter equation based on a higher-rank Lie algebra. These models generalize the famous six-vertex model, which may be viewed in terms of osculating lattice paths, to ensembles of colored paths. By studying certain partition functions within these models, one may define families of multivariate rational functions (or polynomials) with remarkable algebraic features. In these lectures, we will examine a number of these properties:
(a) Exchange relations under the Hecke algebra;
(b) Infinite summation identities of Cauchy-type;
(c) Orthogonality with respect to torus scalar products;
(d) Multiplication rules (combinatorial formulae for structure constants).

Our aim will be to show that all such properties arise very naturally within the algebraic framework provided by the vertex models. If time permits, applications to probability theory will be surveyed.

The magnetohydrodynamics (1 + 1) dimension equation, with a force and force-free term, is analysed with respect to its point symmetries. Interestingly, it reduces to an Abel’s Equation of the second kind and, under certain conditions, to equations specified in Gambier’s family. The symmetry analysis for the force-free term leads to Euler’s Equation and to a system of reduced second-order odes for which singularity analysis is performed to determine their integrability.
 

In this talk I will try to describe how the interplay between the system environment coupling and external driving frequency shapes the dynamical properties and steady state behavior in a periodically driven transverse field Ising chain subject to measurement. I will describe fate of the steady state entanglement scaling properties as a result of measurement induced phase transition. I will briefly explain how such steady state entanglement scaling can be exactly computed using asymptotic analysis of the determinant of associated correlation matrix which turned out to be of block Toeplitz form. I will try to point out the differences from the Hermitian systems in understanding entanglement scaling behaviour in regimes where the asymptotic analysis can be performed using Fisher-Hartwig conjecture. I will end the talk with some open questions in this direction.

Alternating sign matrices (ASM) and descending plane partitions (DPP) both have the concept of rank, and it has been known that the same number of them exist for each rank (conjectured in 1983 by W. H. Mills, David P. Robbins and Howard Rumsey, Jr., and proved in 1996 by Doron Zeilberger and by Greg Kuperberg independently). However, no explicit bijections between them have been found so far. This problem is known as the ASM-DPP bijection problem.

In 2020, Fischer and Konvalinka constructed a bijection between ASM(n)xDPP(n-1) and ASM(n-1)xDPP(n), where ASM(n) denotes the set of ASMs with rank n, and it is similar for DPP(n). This bijection was developed using the concept of signed bijections. I introduce the notion of compatibility of signed bijections to measure the naturalness of signed bijections and to simplify the construction. In this talk, I present the definition of compatibility and some of the results obtained from it. For example, these include the refined structure of Ge...

Totally positive (TP) matrices are matrices in which each minor is positive. First introduced in 1930's by I. Schoenberg and F. Gantmakher and M. Krein, these matrices proved to be important  in many areas of pure and applied mathematics. The notion of total positivity was generalized by G. Lusztig in the context of reductive Lie groups and inspired the discovery  of cluster algebras by S. Fomin and A. Zelevinsky.

In this mini-course, I will first review some basic features of TP matrices, including their spectral properties and discuss some of their classical applications. Then I will focus on weighted networks parametrization of TP matrices due to A. Berenstein, S. Fomin and A. Zelevinsky. I will show how elementary transformations of planar networks lead to criteria of total positivity and important examples of mutations in the theory of cluster algebras. Finally, I will explain how particular sequences of mutations can be used to construct exactly solvable nonlinear dynamical sy...

It is well known that the box-ball system discovered by Takahashi and Satsuma can be obtained by the ultra-discrete analogue of the discrete integrable system, including both the ultra-discrete analogue of the KdV lattice and the ultra-discrete analogue of the Toda lattice. This mini-course will demonstrate that it is possible to derive extended models of the box-ball systems related to the relativistic Toda lattice and the fundamental Toda orbits, which are obtained from the theory of orthogonal polynomials and their extensions. We will first introduce an elementary procedure for deriving box-ball systems from discrete KP equations. Then, we will discuss the relationship between discrete Toda lattices and their extensions based on orthogonal polynomial theory, and outline the exact solutions and ultra-discretization procedures for these systems. Additionally, we will introduce the box-ball system on R, which is obtained by clarifying its relationship with the Pitman transformation in ...