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

For the unharmonic oscillator equation I will show how discriminants, i.e. locus of degenerate eigenvalues, can be plotted inside stability space, literally a space of framed quadratic differentials. They respect chamber structure.

We show that the partial sums of the long Plucker relations for pairs of weakly separated Plucker coordinates oscillate around 0 on the totally nonnegative part of the Grassmannian. Our result generalizes the classical oscillating inequalities by Gantmacher–Krein (1941) and recent results on totally nonnegative matrix inequalities by Fallat–Vishwakarma (2023). In fact we obtain a characterization of weak separability, by showing that no other pair of Plucker coordinates satisfies this property. Weakly separated sets were initially introduced by Leclerc and Zelevinsky and are closely connected with the cluster algebra of the Grassmannian. Moreover, our work connects several fundamental objects such as weak separability, Temperley–Lieb immanants, and Plucker relations, and provides a very general and natural class of additive determinantal inequalities on the totally nonnegative part of the Grassmannian. This is joint work with Soskin.

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...

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.

Combinatorics has constantly evolved from the mere counting of classes of objects to the study of their underlying algebraic or analytic properties, such as symmetries or deformations. This was fostered by interactions with in particular statistical physics, where the objects in the class form a statistical ensemble, where each realization comes with some probability. Integrable systems form a special subclass: that of systems with sufficiently many symmetries to be amenable to exact solutions. In this talk, we explore various basic combinatorial problems involving discrete surfaces, dimer models of cluster algebra, or two-dimensional vertex models, whose (discrete or continuum) integrability manifests itself in different manners: commuting operators, conservation laws, flat connections, quantum Yang-Baxter equation, etc. All lead to often simple and beautiful exact solutions.

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 ...

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 ...

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...

In our study, we model the heart's biological system using percolation on a 2D lattice with three types of cells: Active, Waiting, and Inactive. The system incorporates inhibitory and refractory effects, influenced by two parameters: p_{act}, the probability of a Waiting cell becoming Active, and p_{switch}, the probability of an Inactive cell becoming Waiting. Inactive cells undergo a refractory period before reverting to Waiting.

Our findings show that inhibition raises the percolation threshold, slowing signal propagation. Conversely, reducing the refractory time lowers the threshold and speeds up signal transmission, but it can also trap the signal within the system. We analyzed the distribution of Inactive cells and critical exponents, observing that the Rushbrooke inequality is satisfied.

We consider \emph{bond percolation games} on the $2$-dimensional square lattice in which each edge (that is either between the sites $(x,y)$ and $(x+1,y)$ or between the sites $(x,y)$ and $(x,y+1)$, for all $(x,y) \in \mathbb{Z}^{2}$) has been assigned, \emph{independently}, a label that reads \emph{trap} with probability $p$, \emph{target} with probability $q$, and \emph{open} with probability $1-p-q$. Once a realization of this labeling is generated, it is revealed in its entirety to the players before the game starts. The game involves a single token, initially placed at the origin, and two players who take turns to make \emph{moves}. A \emph{move} involves relocating the token from where it is currently located, say the site $(x,y)$, to any one of $(x+1,y)$ and$(x,y+1)$. A player wins if she is able to move the token along an edge labeled a target, or if she is able to force her opponent to move the token along an edge labeled a trap. The game is said to result in a draw if it cont...

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...