Solvable Lattice Models and Interacting Particle Systems (2023)

Exactly solvable lattice models and interacting particle systems have played a central role in the understanding of universal phenomena and the unification of disparate subjects within the fields of probability, combinatorics and mathematical physics. Exact solvability stems from integrable structures such as solutions to the Yang-Baxter equation and associated special families of symmetric functions (and their non-symmetric analogs). The universal phenomena accessed through these methods provide precise descriptions of the scaling limits of large families of stochastic models spanning from random matrix ensembles, random tilings, random growth processes, directed polymers in random media and many other examples. Though quite different a priori, these models are unified through the common integrable structures into which they fit and through their common scaling limits. These structures also connect disparate areas of mathematics such as p-adic groups, Hecke algebras and cluster algebras to probability theory in novel and useful ways.

The week kicked off on Monday with Vadim Gorin’s talk detailing novel scaling limits of the famous six-vertex model at the boundary of its domain. The six-vertex model and its stochastic specialization, as well as higher spin and higher rank generalizations featured prominently in most of the talks that followed. Higher spin and rank here refer to the capacity of vertices to receive multiple arrows per edge with distinct species or colors. Vertex weights, as in the six-vertex model, are chosen specifically to satisfy the Yang-Baxter equation. James Martin’s talk introduced a central challenge — to understand the nature of stationary measures for stochastic vertex models. Martin focused on periodic boundary conditions and certain higher spin and rank models wherein the stationary measures have remarkable combinatorial interpretations in terms of special variants of Macdonald symmetric polynomials.

Understanding the impact of boundary conditions on stochastic vertex models was a recurrent theme in the symposium and the combined talks of Michael Wheeler and Jan de Gier introduced half-space boundary conditions — in particular, for the asymmetric simple exclusion process, a limit of the stochastic six-vertex model. The presence of such boundary conditions changes both the physical nature of interacting particle systems as well as the integrable structure. This talk focused on new types of formulas for transition probabilities that may provide routes to study physically relevant asymptotics. Combining the study of stationary measures (from Martin’s talk) and boundary conditions (from Wheeler and de Gier’s talks), Guillaume Barraquand’s Tuesday morning talk reported on new methods essentially based on the Yang-Baxter equation for constructing explicit descriptions for the stationary measures for stochastic vertex models with boundary conditions in a half-space and a strip.

Richard Kenyon and Amol Aggarwal’s talk both focused on higher rank models — Kenyon’s on dimer models and their combinatorial properties and Aggarwal’s on models with a mixture of Bosonic and Fermionic species of particles. Aggarwal’s talk introduced a valuable perspective — that many special families of symmetric (and non-symmetric) polynomials can be unified as partition functions of vertex models and thus studied using tools like the Yang-Baxter equation. Leo Petrov’s talk focused on a different manifestation of the Yang-Baxter equation, namely its implications in terms of special symmetries of interacting particle systems. This included a remarkable construction of nice Markov dynamics that reverse time for many classical non-reversible interacting particle systems.

The hike on Wednesday morning (in addition to the other recreational breaks) provided a great opportunity for informal conversations and ruminations around the new directions highlighted in talks. Returning refreshed from these activities, Wednesday afternoon featured two talks. Philippe Di Francesco’s talk focused on combinatorial bijections and outstanding problems in that domain, as well as the tangent method for determining arctic curves for vertex models. Travis Scrimshaw described a theory utilizing objects from K-theory (Grothendieck polynomials) to capture transition probabilities for interacting particle systems in the spirit of the totally asymmetric simple exclusion process. Milind Hedge also delivered an impromptu presentation during the lunch break of work in progress leveraging the higher rank Yang-Baxter equation in order to access asymptotics for height functions of coupled asymmetric simple exclusion processes.

Rinat Kedem’s talk Thursday morning picked up again on the theme of boundary conditions, this time from the perspective of understanding the integrable systems structure of symmetric polynomials (i.e., Koornwinder polynomials and their q-Whittaker type limits) related to root systems with boundary conditions (non-A-type). Jeremy Quastel’s talk focused on the interplay between integrable systems and probability, specifically demonstrating how transition probabilities for the polynuclear growth (PNG) model satisfy the 2D Toda lattice and its non-Abelian generalizations. This integrable structure survives a limit to the universal Kardar-Parisi-Zhang fixed point where the Toda lattice is replaced by the Kadomtsev-Petviashvili equation. Remarkably and as of yet inexplicably, certain aspects of this integrability also seemed to come from the Yang-Baxter equation in Petrov’s earlier talk.

Thursday afternoon’s open problem session was primarily led by Lauren Williams and Sylvie Corteel, who laid out a number of combinatorial mysteries and observations in need of explanation. For instance, Williams described a question about nice factorization formulas for the stationary measure of an inhomogeneous totally asymmetric simple exclusion process in terms of Schubert polynomials. Corteel discussed a novel interactive dimer model originating from the LLT (Leclerc-Lascoux-Thibon) symmetric functions and appealing conjectures about its asymptotic behavior motivated by numerical simulations. The discussion emphasized close ties between algebraic and probabilistic aspects of the subject.

Valentin Buciumas and Jimmy He delivered the Thursday evening talks. Buciumas described combinatorial interpretations of the Casselman-Shalika formula from p-adic representation theory and its generalizations to metaplectic covers of p-adic groups. Remarkably, Buciumas showed how the vertex model approach described earlier in Aggarwal’s talk provides a new approach to such representation theoretic results. He’s talk pivoted back to the question of understanding the influence of boundary conditions on interacting particle systems and vertex models. In particular, it provided a route to study asymptotics of the half-space asymmetric simple exclusion process based on a combination of the usual Yang-Baxter equation and its boundary reflection Sklyanin variant.

Friday started with a talk by Matteo Mucciconi who described a route to extract the large deviation rate function for certain limits of stochastic vertex models. A key innovation highlighted in the talk was the novel use of log-concavity of certain symmetric functions in order to extract the rate function from a related variational problem. Siddhartha Sahi’s probed properties of interpolation polynomials with an aim of prompting the probabilistically inclined audience towards finding applications of these polynomials. Patrik Ferrari’s talk returned to the probabilistic side, focusing on physical phenomena (provable via integrable techniques featured in other talks) about space-time correlations in interacting particle systems. The final talk of the symposium was delivered by Alexey Bufetov. It described higher rank versions of the blocking measures (so-called q-exchangeable measures on the infinite symmetric group) for the asymmetric simple exclusion process and then demonstrated how this model and many others can be seen as random walks on Hecke algebras. Through this lens Bufetov described how many remarkable identities and symmetries emerge.

By bringing together mathematicians and physicists unified in their interest in lattice models but from disparate domains, and through the combination of enlightening talks, exciting open problems and extensive discussions, this symposium has set a key direction of research going forward — understanding and expanding “boundaries.” The first manifestation of this theme is the challenge of uncovering the impact, both in terms of physics and integrable structures, of imposing various sorts of boundary conditions on lattice models and interacting particle systems. The second is a glimmer of new connections between lattice models and domains of mathematics often seen outside the realm of probability and mathematical physics, such as those illustrated in developments related to p-adic representation theory and K-theory.

Amol Aggarwal
Columbia University

Fusion and LLT Polynomials
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Amol Aggarwal will describe a vertex model for the Lascoux-Leclerc-Thibon (LLT) polynomials, whose weights are obtained from fusing the fundamental representations of the quantum affine superalgebra \(U_q (\hat{sl} (1|n))\). Amol Aggarwal will further explain how the structure of the latter can be used to both rederive old and prove new properties of LLT polynomials. This is joint work with Alexei Borodin and Michael Wheeler.
 

Guillaume Barraquand
École Normale Supérieure

Stationary Measures for Integrable Polymers on a Strip
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The solvability of various directed polymer models can be understood through the structures of Schur and Macdonald processes. The definition of these probability measures can be adapted to study models in a half-space as well. However, there does not exist a similar structure for models with two-sided boundaries, i.e., polymers models on a strip. Guillaume Barraquand will show that by viewing the Schur and Macdonald processes as Gibbsian line ensembles, there is a natural way to construct a two-boundary analogue, which allows to determine the stationary measures of last passage percolation, the log-gamma polymer and the KPZ equation on a strip. This is a joint work with Ivan Corwin and Zongrui Yang.
 

Valentin Buciumas
Universiteit van Amsterdam

p-adic Groups, Hecke Algebras and Solvable Lattice Models
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Valentin Buciumas will present some results of interest in the theory of p-adic groups, namely the Satake isomorphism and the (geometric) Casselman-Shalika formula and explain how these results can be interpreted combinatorially. Buciumas will then explain generalizations of these results to metaplectic covers of p-adic groups due to McNamara/Savin and Buciumas-Patnaik.

Finally, Buciumas will explain how results of Aggarwal-Borodin-Wheeler on solvable lattice models (combined with the work of Brubaker-Buciumas-Bump-Gustafsson and observations of Aggarwal) give an alternate proof of the geometric version of the metaplectic Casselman-Shalika formula.
 

Alexey Bufetov
Leipzig University

Interacting Particle Systems and Random Walks on Hecke Algebra
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Multi-species versions of several interacting particle systems, including asymmetric simple exclusion process (ASEP), can be interpreted as random walks on Hecke algebras. In this talk, Alexey Bufetov will discuss this connection and its probabilistic applications to asymptotic results about the convergence of various versions of ASEP.
 

Philippe Di Francesco
University of Illinois, Urbana-Champaign

Arctic Curves for Vertex Models
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Two-dimensional integrable lattice models that can be described in terms of (non-intersecting, possibly osculating) paths with suitable boundary conditions display the arctic phenomenon: the emergence of a sharp phase boundary between ordered crystalline phases (typically near the boundaries) and disordered liquid phases (away from them). Philippe Di Francesco will show how the so-called tangent method can be applied to models such as the 6-vertex model or its triangular lattice variation the 20-vertex model, to predict exact arctic curves. A number of companion combinatorial results are obtained, relating these problems to tiling problems of associated domains of the plane.
 

Vadim Gorin
University of Wisconsin-Madison

Boundary Limits for the Six-Vertex Model
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Take a random configuration of (a,b,c)-weighted six-vertex model in a very large planar domain. What does it look like near a straight segment of the boundary? We investigate this question on the example of the model in N*N square with domain wall boundary conditions and find that the answer depends on the value of \(\Delta=(a^2+b^2-c^2)/(2ab)\): there is a single universal limiting object for all \(\Delta<1\) and a richer class of limits at \(\Delta>1\).
 

Patrik Ferrari
Universität Bonn

On the Space-Time Correlations in KPZ Growth Models
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Patrik Ferrari will discuss results on the correlation structure of some models in the KPZ universality class, for instance, on the covariance of limiting the processes along characteristic lines and of the limiting spatial processes.
 

Jimmy He
Massachusetts Institute of Technology

Boundary Current Fluctuations for the Half Space ASEP
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The half space asymmetric simple exclusion process (ASEP) is an interacting particle system on the half line, with particles allowed to enter/exit at the boundary. Jimmy He will discuss recent work on understanding fluctuations for the number of particles in the half space ASEP started with no particles, which exhibits the Baik-Rains phase transition between Gaussian symplectic ensemble (GSE), Gaussian orthogonal ensemble (GOE) and Gaussian fluctuations as the boundary rates vary. As part of the proof, He will show new distributional identities relating this system to two other models, the half space Hall-Littlewood process, and the free boundary Schur process, which allows exact formulas to be computed.
 

Rinat Kedem
University of Illinois, Urbana-Champaign

Integrability of Quantum Q-systems from Koornwinder Theory
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Rinat Kedem will explain how the specializations of Koornwinder theory, in the q-Whittaker limit, is related to the functional representation of the quantum Q-systems for each of the classical affine root systems. Kedem will show that the conserved quantities, obtained by duality, aka bispectrality, are q-deformed relativistic Toda-type Hamiltonians.
 

Richard Kenyon
Yale University

Higher-Rank Dimers
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On a planar bipartite graph, with edges directed from black to white, associate a vector space with each vertex (possibly of varying dimension) and a linear map for each edge.
Richard Kenyon will show that the associated Kasteleyn matrix determinant is the sum of “traces” of multi-dimer covers (multiwebs).

As a special case, one can study \(GL_n\)-local systems on \(G\), where a multiweb is an n-fold dimer cover. Other special cases include the free-fermionic subvarieties of the 6-vertex, 20-vertex and other vertex models.

This is based on joint work with Dan Douglas, Haolin Shi and Nick Ovenhouse.
 

James Martin
Oxford University

Multi-Type Particle Systems and Interchangeability of Rates
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James Martin will discuss multi-type versions of particle systems, such as the PushTASEP or the totally asymmetric zero-range process, on a ring with site-dependent rates. For example, the stationary weights of a t-deformed multi-type PushTASEP are given by the “ASEP polynomials” (or “permuted basement Macdonald polynomials”) studied recently, e.g., by Chen, de Gier and Wheeler, by Corteel, Mandelshtam and Williams and by Alexandersson.

The links to families of symmetric functions lead to symmetry properties of the stationary distribution under interchange of the rates at different sites. In some cases, there are probabilistic arguments which give stronger interchange properties, applying to evolutions of the process over time, both in and out of stationarity. Martin conjectures that these pathwise interchangeability properties hold more widely — many open questions remain.
The talk is based on joint work with Arvind Ayyer, Olya Mandelshtam and Omer Angel.
 

Matteo Mucciconi
University of Warwick

Large Deviations for the Height Function of the q-Deformed Polynuclear Growth
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The q-deformed polynuclear growth is a growth process that generalizes the polynuclear growth studied in the context of KPZ universality class. In this talk, Matteo Mucciconi will discuss the mathematical derivation of large time deviations for the height function. Rare events, as functions of the time t, display distinct decay rates based on whether the height function grows significantly larger (upper tail) or smaller (lower tail) than the expected value. Upper tails exhibit an exponential decay \(e^{−tΦ+(·)}\) with rate function \(Φ+(·)\) which we determine explicitly. Conversely, the lower tails experience a more rapid decay as \(e^{−t^2 Φ−(·)}\) and the rate function \(Φ−(·)\) is given in terms of a variational problem.
 

Leo Petrov
University of Virginia

Stochastic Applications of the Yang-Baxter Equation
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The Yang-Baxter equation (YBE) is a celebrated tool in quantum integrable systems. Leo Petrov will discuss its recent stochastic interpretations and applications, including refinement and randomization of the YBE for the six-vertex model and its higher spin variant. This leads to a unified treatment of Cauchy identities for various families of symmetric functions (together with RSK-like randomized bijections) and monotone couplings in interacting particle systems with varying speed parameters. Moreover, stationary measures of colored stochastic vertex models on the line and the circle can also be derived directly from the YBE (previously, they were constructed using matrix product ansatz or multiline queues).
 

Jeremy Quastel
University of Toronto

PNG/Toda
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The Kolmogorov backward equation for the polynuclear growth model is solved in a fairly transparent way with Fredholm determinants. One-point distributions come from the 2d Toda lattice; multipoints, the non-Abelian Toda lattice. These scale readily to Fredholm determinants and KP equations for the KPZ fixed point. Joint work with Konstantin Matetski and Daniel Remenik.
 

Siddhartha Sahi
Rutgers University

Interpolation Polynomials, Bar Monomials and Their Positivity

The interpolation polynomials are a remarkable family of inhomogeneous multivariate polynomials, whose top homogeneous parts are Jack polynomials, and which are characterized by certain elementary vanishing properties.
Siddhartha Sahi will describe the proof of a positivity result for interpolation polynomials, which was obtained recently together with Yusra Naqvi and Emily Sergel. This result was conjectured in the mid-1990s by F. Knop and the speaker; it is a natural generalization of Macdonald’s positivity conjecture for Jack polynomials, which was proved by them around the same time.
The main tool is an inhomogeneous analog of an ordinary monomial called a bar monomial, and whose positivity we prove via the combinatorics of compositions. This involves a new partial order that is called the bar order, and a new operation that is called a glissade.
 

Travis Scrimshaw
Hokkaido University

Lattice Models for Refined Grothendieck Polynomials and TASEP
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Schubert calculus is a now classical subject in algebraic combinatorics arising from the cohomology of the flag variety and Grassmannian. A more modern study is about the corresponding K-theoretic objects, with refined parameters coming from the combinatorial descriptions. In this talk, Travis Scrimshaw will first construct solvable lattice models whose partition functions are the (dual) symmetric Grothendieck polynomials from this combinatorial data. Using this, Scrimshaw will reinterpret this using free fermions and the boson-fermion correspondence. Now there is an algebraic framework to encode that offers a description of the dynamics of the versions of TASEP studied by Dieker and Warren. This is based on joint work with Shinsuke Iwao and Kohei Motegi.
 

Michael Wheeler & Jan de Gier
University of Melbourne

The Asymmetric Exclusion Process on the Half Line with General Open Boundaries, I & II
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In two tandem talks, Michael Wheeler and Jan de Gier will report on the derivation of an explicit expression for the transition probability of the ASEP on the half line with general open boundaries. Using a vertex model approach we derive contour integral expressions for these transition probabilities, with particular emphasis on the empty-to-empty transition, which is the simplest while retaining all non-trivial features of the generic non-diagonal boundaries.