Solvable Lattice Models & Interacting Particle Systems (2025)

Amol Aggarwal
Columbia University

Asymptotics for the Toda Lattice 
View Slides (PDF)

The Toda lattice prescribes the evolution of N particles interacting under certain Hamiltonian dynamics; it is an archetypal example of a completely integrable system. A question of interest is to understand how the model behaves, under random (or typical) initial data, when the number N of particles becomes large. This talk describes several results explaining such asymptotics under certain invariant initial data. The proofs proceed by finding a way to interpret the Toda lattice as a dense collection of “quasi-particles” that behave similarly to solitons, and providing a framework to study how these quasi-particles asymptotically evolve in time. In this analysis, arguments from random matrix theory, particularly the analysis of Lyapunov exponents governing the decay rates of eigenvectors of random tridiagonal matrices, play an important role.
 

Alexey Bufetov
University of Leipzig

Stochastic 3D Twenty-Vertex Model

We consider a stochastic three-dimensional twenty-vertex model. We study its basic properties, such as stationary measures and dualities, and establish the convergence to the telegraph equation in a certain asymptotic regime. This is a joint work with Panagiotis Zografos.
 

Jan de Gier
University of Melbourne

Inhomogeneous q-deformation of the Tsetlin Library
View Slides (PDF)

The Tsetlin library is a random-to-top shuffle process on permutations. A q-deformation using the Hecke algebra can also be implemented on flags over the finite field \()\mathbb{F}_q^n\). The semigroup underlying the random-to-top Markov chain on flags is a left regular band, which provides a method to compute the stationary distribution. We consider a further generalisation by introducing inhomogeneous rates. This is work in collaboration with Arvind Ayyer, Sarah Brauner, and Anne Schilling.
 

Percy Deift
New York University

Applications of a Commutation Formula II
View Slides (PDF)

In 1978, Percy Deift described a variety of applications of the commutation formula λ(λ + AB) ⁻¹ + A(λ + BA) ⁻¹ B = 1 for operators A, B in a Banach space, to problems in mathematics and physics. These problems included isospectrality, KdV, regularization of PDE’s, and operator estimates in statistical mechanics and quantum field theory.

Over the intervening 50 years or so, many other applications of the commutation formula have been recognized. In this lecture, Deift will describe some of these applications in various fields of mathematics and physics and in which commutation plays a key role. These applications include determinants and dimension reduction, random matrix theory and dimension expansion, integrable operators, Poncelet’s porism, interpolation theory without complex analysis, eigenvalue computation, and KPZ.

This is joint work with Fritz Gesztesy.
 

Patrik Ferrari
University of Bonn

Large Time Limit of the Two-Point Function in a Two-Species ASEP
View Slides (PDF)

We consider the two-species partially asymmetric simple exclusion process on Z, starting with translation-invariant stationary measure. We establish the large time limit of the two-point function in the associated normal modes. This confirms the predictions of the nonlinear fluctuating hydrodynamic theory. Joint work with Sabrina Gernholt.
 

Pavel Galashin
University of California, Los Angeles

Amplituhedra and Origami
View Slides (PDF)

Pavel Galashin will explain a proof of the BCFW triangulation conjecture which states that the cells appearing in the Britto–Cachazo–Feng–Witten (BCFW) recursion triangulate the amplituhedron (in full generality at all loop levels). The key ingredient is a relation to origami crease patterns which are planar graphs with faces colored black and white, embedded in the plane so that the sum of black (equivalently, white) angles at each vertex is 180°. Along the way, we prove conjectures of Chelkak–Laslier–Russkikh and Kenyon–Lam–Ramassamy–Russkikh on the existence of such origami embeddings of arbitrary planar graphs, which originated from the works of Kenyon and Smirnov on the conformal invariance of the dimer and Ising models.
 

Vadim Gorin
University of California, Berkeley

Airy-Green Function as a Tool for Uniform Inference for Signal Strength in Signal Plus Noise Models
View Slides (PDF)

We discuss four classical signal-plus-noise models: the sum of a Wigner matrix and a low-rank perturbation, spiked sample covariance matrices, the factor model, and canonical correlation analysis with low-rank dependencies. Our objective is to construct confidence intervals for the signal strength that are uniformly valid across all regimes — strong, weak, and critical signals. We demonstrate that traditional Gaussian approximations fail in the critical regime. Instead, we put forward a universal transitional distribution that enables valid inference across the entire spectrum of signal strengths. The crucial role is played by the (stochastic) Airy–Green function, which we are going to define and examine.
 

Jimmy He
Ohio State University

Symmetries of Periodic and Free Boundary Measures on Partitions
View Slides (PDF)
View Slides (PDF)

The periodic and free boundary q-Whittaker measures are probability measures on partitions defined in terms of q-Whittaker functions and an additional parameter \(u\) controlling the behavior of the system at the boundary. Jimmy He will explain a hidden distributional symmetry of this model which exchanges the \(u\) and \(q\) parameters, as well as related results on Hall-Littlewood measures. As a special case, we recover identities of Imamura–Mucciconi–Sasamoto. This is joint work with Michael Wheeler.
 

Jiaoyang Huang
University of Pennsylvania

A Convergence Framework for the Airy_beta Line Ensemble via Pole Evolution
View Slides (PDF)

The Tracy–Widom beta = 2 distribution is the marginal distribution of the top curve in the Airy line ensemble, which consists of an infinite sequence of random curves introduced by Prähofer and Spohn. This ensemble was conjectured to describe the scaling limit of various random surfaces and stochastic growth models within the Kardar–Parisi–Zhang (KPZ) universality class. More generally, the Tracy-Widom beta distribution represents the marginal distribution of the top curve in the Airy beta line ensemble, which arises from the scaling limit of Dyson’s Brownian motion.

In this talk, we will present a characterization result for the Airy beta line ensemble via a stochastic differential equation and discuss its applications, particularly in the convergence of the Airy beta line ensemble for Dyson Brownian motions with general potentials, Laguerre processes, and Jacobi processes. These results are based on joint work with Lingfu Zhang.
 

Richard Kenyon
Yale University

Kasteleyn’s Theorem for Classical Groups
View Slides (PDF)

Kasteleyn’s theorem shows how to count dimer covers of a planar graph G with the Pfaffian of an associated adjacency-type matrix. For a graph G with connection taking values in SL(n), SO(n), or Sp(2n), we extend Kasteleyn’s theorem to count traces of “multiwebs” in G. This joint work with Nick Ovenhouse and Haihan Wu.
 

Karol Kozlowski
École Normale Supérieure, Lyon

Towards a Rigorous Theory of Bethe Ansatz Equations

Numerous exactly solvable models of \(1+1\) dimensional many-body quantum mechanics or \(2\) dimensional classical statistical mechanics can be solved by means of the Bethe ansatz. More precisely, this technique allows one to construct eigenvectors and eigenvalues of the underlying many-body Hamiltonian or of the associated transfer matrix. Both eigenvectors and eigenvalues are parameterised by implicit solutions to the celebrated Bethe ansatz equations. In the simplest setting, these are coupled equations for \(N\) unknowns \()\la_1,\dots, \la_N \in \Cx\), which take the form

\(\Phi(\lambda_a)^L =\)

\((-1)^{N-1} \prod\limits_{b=1}^{N} \bigg\{ \frac{ \sinh({\mathrm i } \zeta + \lambda_a-\lambda_b ) }{ \sinh( {\mathrm i } \zeta + \lambda_b-\lambda_a ) } \bigg\} \;, \qquad a=1,\dots, N,\)

where \(\zeta\) is related to the interaction strength in the model and the meromorphic function \(\Phi\) depends on the model.

The knowledge of these solutions then opens up the possibility to study the quantities of main physical and mathematical interest: various properties of the spectrum to start with along with the correlation functions encoding all of the interesting properties of such systems, the various facetes of their universal behaviours in particular.

However, the mathematical status of all this construction is still in its infancy, especially from the point of view of analysis aspects. Indeed, one is interested in describing the various mentioned objects in the limit when \(L, N \rightarrow + \infty\) with \(\frac{N}{L} \rightarrow D \in \mathbb{R}^+$\). This demands an understanding of the behaviour of certain classes of solutions to the associated Bethe ansatz equations in the mentioned limit. When \(\zeta=\frac{\pi}{2}\), the so-called free fermion point, the equations decouple and the problem trivialises. For generic \(\zeta\) and when \(D=0\), an appropriate mathematical understanding has also been reached. However, the \(D>0\) regime is swarming with open problems.

In this talk, Karol Kozlowski will describe a method allowing one to rigorously describe, in the limit of interest, a large class of solutions to the Bethe equations given above with \(D>0\). Kozlowski will show how this information allows one to access, on rigorous grounds, some of the universal properties of the spectra of the underlying quantum interacting, { \it viz}. away from the free fermion point, integrable models.
 

Pierre Le Doussal
École Normale Supérieure

Integrability and Exact Large Deviations of the Weakly-Asymmetric Exclusion Process
View Slides (PDF)

We provide a solution to the macroscopic fluctuation theory MFT of the 1D weakly asymmetric exclusion process (WASEP) with two sided stationary initial conditions on the infinite line. We obtain the exact large deviation function of the current, its cumulants, and tracer position. It interpolates between known results for the SSEP and the KPZ equation at weak noise. We show integrability of the MFT of the WASEP by a mapping to a spin model, with explicit Lax matrices.

Based on joint work with Alexandre Krajenbrink and Pierre Le Doussal, arXiv:2505.12034
 

Vladimir Mangazeev
Australia National University

Towards a Q-operator for the Open XXZ Chain
View Slides (PDF)

We study solutions of the reflection equation for the XXZ spin model for arbitrary spins. The standard method of the construction of the Q-operator fails for the case of non-diagonal K-matrices. Motivated by the construction of Lazarescu and Pasquier, we consider a double-row transfer matrix with an infinite-dimensional auxiliary space and fixed boundary conditions. Commutativity with the original transfer matrix produces new local equations for boundary vectors. Using this construction, we can calculate XXZ K-matrices with an arbitrary spin and find their generating function. After applying a special similarity transformation to the L-operator, we compare this result with a previosly known solution of the reflection equation for an integer spin.
 

Kenneth McLaughlin
Tulane University

Asymptotic Analysis of Multisoliton Solutions of Integrable Partial Differential Equations and the Kinetic Theory of Soliton Gasses

The kinetic theory of solitons and soliton gases originates from the discovery of the complete integrability of the Korteweg–de Vries (KdV) equation in the 1960s, leading to the identification of solitons as fundamental nonlinear phenomena. The concept of a soliton gas was introduced by Zakharov in 1971 and further developed by El in 2003, modeling solitons as interacting particle-like structures. In recent years, rigorous analytical results have been established that provide confirmation of the qualitative theory.

In this talk, Kenneth McLaughlin will describe some of them, including (1) a rigorous derivation of kinetic equations governing soliton gases in KdV-type systems without randomness, as well as (2) the analysis of random collections of solitons, in which both mean behavior and fluctuation results are established. This is joint work with several teams, including Manuela Girotti, Aikaterini Gkogkou, Tamara Grava, Robert Jenkins, Guido Mazzuca, Oleksandr Minakov, and Maxim Yattselev.
 

Matteo Mucciconi
National University of Singapore

Elliptic Functions in The Lower Tail of the Q-Deformed Polynuclear Growth

The q-deformed polynuclear growth is a model of a one-dimensional growing interface which interpolates between the celebrated polynuclear growth and the Kardar–Parisi–Zhang equation. Calling h(x,t) the height of the interface at location x and time t, we consider the problem of establishing sharp asymptotics for the probability that h(x,t) assumes values substantially lower than its average. We compute explicitly the lower tail large deviation rate function, which is expressed in a closed form through Weierstrass elliptic functions. The derivation of the rate function relies on the Riemann–Hilbert analysis of a discrete log-gas associated with the Bessel functions. The equilibrium measure of such Riemann–Hilbert problem presents, in general, saturated regions and it undergoes two phase transitions, a phenomenon which might be of independent interest. Based on a joint work with Mattia Cafasso and Giulio Ruzza.
 

Leonid Petrov
University of Virginia

Random Lozenge Waterfall
View Slides (PDF)

Uniformly random lozenge tilings of a hexagon constitute a well-known model that develops limit shapes and exhibits universal fluctuations — local, global, and edge, when the sides of the hexagon are scaled proportionally. The uniform model admits integrable deformations: the q^volume-weighted measure displays much of the same behavior as q->1, or degenerates to a “tropical” shape when q is fixed and the sides grow to infinity. Adding one more parameter kappa leads to the q-Racah model, introduced by Borodin–Gorin–Rains in 2009. Keeping q fixed, one sees a new phenomenon: the appearance of a “waterfall,” which is a flat limit shape not parallel to a coordinate plane, and within which one gets a one-dimensional random stepped interface. Leonid Petrov will discuss our recent progress in establishing the limit shape and conjectures on the point process inside the waterfall.

Based on joint work with Alisa Knizel.
 

Tomaž Prosen
University of Ljubljana

(Deformed) Rule 54: Unusual Integrable Model
View Slides (PDF)

Tomaž Prosen will discuss a fascinating example of exactly solvable reversible cellular automaton in 1+1 dimensions, the rule 54, and two of its deformations which turn the model into either quantum or stochastic cellular automaton. The (deformed) model can be interpreted as arguably the simplest integrable dynamics with interaction round-a-face. Both deformations, being parametrixed by an arbitrary unitary or stochastic 2×2 matrix, respectively, exhibit some features of integrability. The later (stochastic version) can be interpeted as a lattice discretization of deformed t-PNG model. Prosen will discuss the construction of non-trivial conservation laws of the quantum deformation and the matrix product form of the steady state of the boundary driven stochastic deformation. If time permits, Prosen will also mention a certain 2+1 dimensional deterministic extention exibiting robust soliton scattering and rich manifold of strongly correlated time-translation invariant states.
 

Herbert Spohn
Technical University of Munich

Universality Classes for Coupled Kardar-Parisi-Zhang Equations
View Slides (PDF)

Considered is the KPZ equation with two components, constrained to be symmetric under exchange. This model is of interest in the context of exciton-polaritons condensates. A central issue is to understand the structure of universality classes, which are defined by having the same spacetime correlator up to dilations and rotations. As for the single component, the static scaling exponent is 1/2, and the dynamic scaling exponent seems to be 3/2 according to our numerics. However, recent simulations by Schmidt, et al., indicate two distinct dynamical scaling exponents, each one different from 3/2. This is joint work together with Dipankar Roy, Abhishek Dhar, and Manas Kulkarni.