Solvable Lattice Models and Interacting Particle Systems (2023)
Organizers:
Alexei Borodin, MIT
Ivan Corwin, Columbia University
The Simons Symposium on Solvable Lattice Models and Interacting Particle Systems explored the realm of stochastic integrable lattice models that lies at the intersection of two large subjects of YangBaxter solvable lattice models and Markovian stochastic growth models in (1+1) and (2+1)dimensions. A particular focus was on the role of the colored vertex models, which correspond to quantum affine algebras of higher rank and to multispecies Markovian systems, and on correspondences between freefermionic and nonfreefermionic models. Other directions explored include the relationship between vertex models and symmetric functions, open interacting particle systems and their integrability, and directed polymers in random media.

Meeting Report
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 YangBaxter equation and associated special families of symmetric functions (and their nonsymmetric 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 padic 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 sixvertex model at the boundary of its domain. The sixvertex 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 sixvertex model, are chosen specifically to satisfy the YangBaxter 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 halfspace boundary conditions — in particular, for the asymmetric simple exclusion process, a limit of the stochastic sixvertex 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 YangBaxter equation for constructing explicit descriptions for the stationary measures for stochastic vertex models with boundary conditions in a halfspace 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 nonsymmetric) polynomials can be unified as partition functions of vertex models and thus studied using tools like the YangBaxter equation. Leo Petrov’s talk focused on a different manifestation of the YangBaxter 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 nonreversible 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 Ktheory (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 YangBaxter 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 qWhittaker type limits) related to root systems with boundary conditions (nonAtype). 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 nonAbelian generalizations. This integrable structure survives a limit to the universal KardarParisiZhang fixed point where the Toda lattice is replaced by the KadomtsevPetviashvili equation. Remarkably and as of yet inexplicably, certain aspects of this integrability also seemed to come from the YangBaxter 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 (LeclercLascouxThibon) 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 CasselmanShalika formula from padic representation theory and its generalizations to metaplectic covers of padic 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 halfspace asymmetric simple exclusion process based on a combination of the usual YangBaxter 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 logconcavity 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 spacetime 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 (socalled qexchangeable 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 padic representation theory and Ktheory.

Agenda
MONDAY, JUNE 19
10:00  11:00 AM Vadim Gorin  Boundary Limits for the SixVertex Model 11:30  12:30 PM James Martin  MultiType Particle Systems and Interchangeability of Rates 5:00  6:00 PM Michael Wheeler  The Asymmetric Exclusion Process on the Half Line with General Open Boundaries, I 6:15  7:15 PM Jan de Gier  The Asymmetric Exclusion Process on the Half Line with General Open Boundaries, II TUESDAY, JUNE 20
10:00  11:00 AM Guillaume Barraquand  Stationary Measures for Integrable Polymers on a Strip 11:30  12:30 PM Richard Kenyon  HigherRank Dimers 5:00  6:00 PM Amol Aggarwal  Fusion and LLT Polynomials 6:15  7:15 PM Leo Petrov  Stochastic Applications of the YangBaxter Equation WEDNESDAY, JUNE 21
5:00  6:00 PM Philippe Di Francesco  Arctic Curves for Vertex Models 6:15  7:15 PM Travis Scrimshaw  Lattice Models for Refined Grothendieck Polynomials and TASEP THURSDAY, JUNE 22
10:00  11:00 AM Rinat Kedem  Integrability of Quantum QSystems from Koornwinder Theory 11:30  12:30 PM Jeremy Quastel  PNG/Toda 5:00  6:00 PM Valentin Buciumas  padic Groups, Hecke Algebras and Solvable Lattice Models 6:15  7:15 PM Jimmy He  Boundary Current Fluctuations for the Half Space ASEP FRIDAY, JUNE 23
10:00  11:00 AM Matteo Mucciconi  Large Deviations for the Height Function of the qdeformed Polynuclear Growth 11:30  12:30 PM Siddhartha Sahi  Interpolation Polynomials, Bar Monomials and their Positivity 2:45  3:45 PM Patrik Ferrari  On the SpaceTime Correlations in KPZ Growth Models 3:50  4:50 PM Alexey Bufetov  Interacting Particle Systems and Random Walks on Hecke Algebra 5:00 PM Concert  Mischa Maisky & Lily Maisky 
Abstracts & Slides
Amol Aggarwal
Columbia UniversityFusion and LLT Polynomials
View Slides (PDF)Amol Aggarwal will describe a vertex model for the LascouxLeclercThibon (LLT) polynomials, whose weights are obtained from fusing the fundamental representations of the quantum affine superalgebra \(U_q (\hat{sl} (1n))\). 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 SupérieureStationary Measures for Integrable Polymers on a Strip
View Slides (PDF)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 halfspace as well. However, there does not exist a similar structure for models with twosided 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 twoboundary analogue, which allows to determine the stationary measures of last passage percolation, the loggamma polymer and the KPZ equation on a strip. This is a joint work with Ivan Corwin and Zongrui Yang.
Valentin Buciumas
Universiteit van Amsterdampadic Groups, Hecke Algebras and Solvable Lattice Models
View Slides (PDF)Valentin Buciumas will present some results of interest in the theory of padic groups, namely the Satake isomorphism and the (geometric) CasselmanShalika formula and explain how these results can be interpreted combinatorially. Buciumas will then explain generalizations of these results to metaplectic covers of padic groups due to McNamara/Savin and BuciumasPatnaik.
Finally, Buciumas will explain how results of AggarwalBorodinWheeler on solvable lattice models (combined with the work of BrubakerBuciumasBumpGustafsson and observations of Aggarwal) give an alternate proof of the geometric version of the metaplectic CasselmanShalika formula.
Alexey Bufetov
Leipzig UniversityInteracting Particle Systems and Random Walks on Hecke Algebra
View Slides (PDF)Multispecies 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, UrbanaChampaignArctic Curves for Vertex Models
View Slides (PDF)Twodimensional integrable lattice models that can be described in terms of (nonintersecting, 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 socalled tangent method can be applied to models such as the 6vertex model or its triangular lattice variation the 20vertex 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 WisconsinMadisonBoundary Limits for the SixVertex Model
View Slides (PDF)Take a random configuration of (a,b,c)weighted sixvertex 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 \(\Delta=(a^2+b^2c^2)/(2ab)\): there is a single universal limiting object for all \(\Delta<1\) and a richer class of limits at \(\Delta>1\).
Patrik Ferrari
Universität BonnOn the SpaceTime Correlations in KPZ Growth Models
View Slides (PDF)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 TechnologyBoundary Current Fluctuations for the Half Space ASEP
View Slides (PDF)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 BaikRains 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 HallLittlewood process, and the free boundary Schur process, which allows exact formulas to be computed.
Rinat Kedem
University of Illinois, UrbanaChampaignIntegrability of Quantum Qsystems from Koornwinder Theory
View Slides (PDF)Rinat Kedem will explain how the specializations of Koornwinder theory, in the qWhittaker limit, is related to the functional representation of the quantum Qsystems for each of the classical affine root systems. Kedem will show that the conserved quantities, obtained by duality, aka bispectrality, are qdeformed relativistic Todatype Hamiltonians.
Richard Kenyon
Yale UniversityHigherRank Dimers
View Slides (PDF)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 multidimer covers (multiwebs).As a special case, one can study \(GL_n\)local systems on \(G\), where a multiweb is an nfold dimer cover. Other special cases include the freefermionic subvarieties of the 6vertex, 20vertex and other vertex models.
This is based on joint work with Dan Douglas, Haolin Shi and Nick Ovenhouse.
James Martin
Oxford UniversityMultiType Particle Systems and Interchangeability of Rates
View Slides (PDF)James Martin will discuss multitype versions of particle systems, such as the PushTASEP or the totally asymmetric zerorange process, on a ring with sitedependent rates. For example, the stationary weights of a tdeformed multitype 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 WarwickLarge Deviations for the Height Function of the qDeformed Polynuclear Growth
View Slides (PDF)The qdeformed 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 VirginiaStochastic Applications of the YangBaxter Equation
View Slides (PDF)The YangBaxter 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 sixvertex model and its higher spin variant. This leads to a unified treatment of Cauchy identities for various families of symmetric functions (together with RSKlike 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 TorontoPNG/Toda
View Slides (PDF)The Kolmogorov backward equation for the polynuclear growth model is solved in a fairly transparent way with Fredholm determinants. Onepoint distributions come from the 2d Toda lattice; multipoints, the nonAbelian 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 UniversityInterpolation 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 mid1990s 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 UniversityLattice Models for Refined Grothendieck Polynomials and TASEP
View Slides (PDF)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 Ktheoretic 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 bosonfermion 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 MelbourneThe Asymmetric Exclusion Process on the Half Line with General Open Boundaries, I & II
View Wheeler Slides (PDF)
View de Gier Slides (PDF)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 emptytoempty transition, which is the simplest while retaining all nontrivial features of the generic nondiagonal boundaries.