ICTS

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 probability theory. Furthermore, we will also discuss how extended box-ball systems can compute the Smith normal form of bidiagonal integer matrices.

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 continues indefinitely (i.e.\ with the token always being moved along open edges). We ask the question: for what values of $p$ and $q$ is the probability of draw equal to $0$? By establishing a close connection between the event of draw and the \emph{ergodicity} of a suitably defined \emph{probabilistic cellular automaton}, we are able to show that the probability of draw is $0$ when $p > 0.157175$ and $q=0$, and when $p=q \geqslant 0.10883$

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 probability theory. Furthermore, we will also discuss how extended box-ball systems can compute the Smith normal form of bidiagonal integer matrices.

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 probability theory. Furthermore, we will also discuss how extended box-ball systems can compute the Smith normal form of bidiagonal integer matrices.

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.

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.

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).

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.

In this talk, we consider the skein algebra (or cluster algebra) of an oriented marked surface with boundary. Each triangulation of the surface determines an algebraic torus in the corresponding variety, and in some cases these tori collectively cover the variety. In other cases, there are points not contained in any algebraic torus, which we call deep points. In joint work with Greg Muller, we have classified the deep points of skein algebras of unpunctured marked surfaces. We will discuss this classification, beginning with convex polygons and then extending those same ideas to other unpunctured surfaces.

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 probability theory. Furthermore, we will also discuss how extended box-ball systems can compute the Smith normal form of bidiagonal integer matrices.

A system of hard rigid rods of length $k \gg1$ on hypercubic lattices in dimensions $d \geq2$, is known to undergo two phase transitions when chemical potential is increased: from a low-density phase to an intermediate density nematic phase, and on further increase to a high-density phase with no nematic order. I will present non-rigorous arguments to support the conjecture that for large $k$, the second phase transition is a first-order transition with a discontinuity in density in all dimensions greater than $1$. The chemical potential at the transition is $\approx A k \ln k$ for large $k$, and that the density of uncovered sites drops from a value $\approx B (\ln k)/ k^2$ , to a value of order $\exp(−ck)$, where $c$ is some constant, across the transition. We conjecture that these results are asymptotically exact, and $A = B= 1$, in all dimensions $d ≥ 2$.

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-order ones. This is joint work with A. Ramani, R. Willox and T. Mase.

I will discuss the relation between non-compact spin chains and the zero-range processes introduced by Sasamoto-Wadati, Povolotsky and Barraquand-Corwin. The main difference compared to the prime examples of integrable particle processes, namely the SSEP and the ASEP, is that for the models discussed in this talk several particles can occupy one and the same site. Guided by the desire to maintain the integrable structure, I will introduce boundary conditions for these models that are obtained from the boundary Yang-Baxter equation. This allows to define analogues of the open SSEP and ASEP with boundary reservoirs. Some recent exact results concerning these types of integrable non-equilibrium zero-range processes will be discussed.

In this talk we shall present different approaches to the direct and inverse spectral problems for a family of solutions of KP2 , highlight the interplay of tropical geometry and combinatorics in solving such problems, and present unexpected connections to other problems in theoretical physics and statistical mechanics.

We propose a four-dimensional analogue of a multiplicative QRT mapping and construct its space of initial conditions as a rational variety of which the mapping becomes a pseudo-automorphism. We embed this variety into a family which admits symmetries forming an extended affine Weyl group and use this to construct a non-autonomous version of the mapping which can be regarded as a fourth order $q$ Painlevé equation. We also construct another $q$-Painlevé type mapping from a different translation element of the symmetry group.
Based on joint work with Adrian Stefan Carstea and Tomoyuki Takenawa.

In [1] we introduced the skew RSK dynamics, which is a time evolution for a pair of skew Young tableaux (P,Q). This gives a connection between the q-Whittaker measure and the periodic Schur measure, which immediately implies a Fredholm determinant formula for various KPZ models[2]. The dynamics exhibits interesting solitonic behaviors similar to box ball systems (BBS) and is related to the theory of crystal. 

In this talk we explain basics of the skew RSK dynamics. The talk is based on a collaboration with T. Imamura, M. Mucciconi. 

[1] T. Imamura, M. Mucciconi, T. Sasamoto, 
Skew RSK dynamics: Greene invariants, affine crystals and applications to $q$-Whittaker polynomials, Forum of Mathematics, Pi (2023), e27 1–10. 

[2] T. Imamura, M. Mucciconi, T. Sasamoto, 
Solvable models in the KPZ, arXiv: 2204.08420

I will first present a generic argument to derive large deviations of a stochastic process when large deviations of certain functionals of that process are available. I will then apply such a general argument to the analysis of the lower tail of the height functions of the stochastic six vertex model starting with step initial conditions. One of the main novelties will be a proof of weak logarithmic concavity of the cumulative distribution function of the height function. This is a joint work with Sayan Das and Yuchen Liao.

The quantum K-theory of the flag variety is a ring defined by introducing a quantum product to the K-theory of the flag variety. Under appropriate localization, it is known that the following three rings (i), (ii), and (iii) are isomorphic, and this property allows for a detailed investigation of each ring: (i)the coordinate ring of the phase space of the relativistic Toda lattice, (ii) the quantum equivariant K-theory of the flag variety, and (iii) the K-equivariant homology ring of the affine Grassmannian.

The isomorphism between (i) and (ii) is derived from the Lax formalism of the relativistic Toda lattice [Ikeda-Iwao-Maeno]. The isomorphism between (ii) and (iii) is referred to as the K-Peterson isomorphism [Lam-Li-Mihalcea-Shimozono, Kato, Chow-Leung, Ikeda-Iwao-Maeno]. In this talk, I will outline how techniques from classical integrable systems, such as the construction of algebraic solutions and Bäcklund transformations, are applied to the study of geometry. This talk is based on a joint work with Takeshi Ikeda (Waseda University), Satoshi Naito (Tokyo Institute of Technology / Institute of Science Tokyo), and Kohei Yamaguchi (Nagoya University).

In this talk, I'll describe some recently discovered connections between one-dimensional interacting particle models (the ASEP and the TAZRP) and Macdonald polynomials and show the combinatorial objects that make these connections explicit. Recently, a new tableau formula was found for the modified Macdonald polynomial $\widetilde{H}_{\lambda}$ in terms of a queue inversion statistic that is naturally related to the dynamics of the TAZRP. We give a new compact tableau formula for the symmetric Macdonald polynomials $P_{\lambda}(X;q,t)$ using the same queue inversion statistic on certain sorted non-attacking tableaux. The nonsymmetric components of our formula are the ASEP polynomials, which specialize to the probabilities of the asymmetric simple exclusion process (ASEP) on a circle, and the queue inversion statistic encodes to the dynamics of the ASEP.  Our tableaux are in bijection with Martin's multiline queues, from which we obtain an alternative multiline queue formula for $P_{\lambda}$.