Colored vertex model

All roads to the KPZ universality class

Workshop at American Institute of Mathematics, Pasadena, CA

March 17-21, 2025

Organized by Axel Saenz (Oregon State) and Leonid Petrov (UVA)

Participant contributions

Kailun Chen

Statement for “All roads to the KPZ universality class”

Convergence of ASEP speed process to KPZ stationary horizon

The KPZ stationary horizon was first introduced in [Bus24] and is expected to be a universal scaling limit for multi-class invariant measures of models in the KPZ universality class. Then [BSS22] shows that the suitably rescaled TASEP speed process converges weakly to the KPZ stationary horizon. Recently, [BSS24] developed a more general framework to show convergence to the stationary horizon. In particular, they show that if a model converges to the directed landscape under suitable rescaling, then the stationary measures of the associated multi-class process converge to the stationary horizon at the level of finite-dimensional projections. In [ACH24a] they prove the convergence of the stochastic six-vertex model and ASEP to the directed landscape, and hence, using the results from [BSS24], they obtain as a corollary that the stationary measures for the multi-class ASEP converge to marginals of the stationary horizon.

The ASEP speed process and stochastic six-vertex model speed process were recently introduced in [ACG23] and [DHS24] separately. It is an open problem to prove the convergence of the ASEP and stochastic six-vertex model speed processes to the KPZ stationary horizon.

Integrable structure of ASEP and KPZ fixed point

The KPZ fixed point was first introduced in [MQR21] by using exact determinantal formulas for the totally asymmetric simple exclusion process (TASEP). Recently, [QS23] and [ACH24b] use “soft” methods to prove the convergence of ASEP to the KPZ fixed point. However, we know little about the integrable structure of ASEP. It is an open problem to prove the convergence of ASEP to the KPZ fixed point by using exact formulas.

References:

Nikolaus Elsaesser

My interests in the workshop span questions regarding when and why certain combinatorial identities appear in exactly solved models in the KPZ universality class, what kinds of gaps exist between the mathematics and experiments in KPZ research, the potential applications/use cases of research in KPZ universality to experiments and other areas of mathematics, and what steps are needed to realize said applications.

Milind Hegde

The role of Yang–Baxter integrability in universality; connections between such integrability and other forms like the RSK correspondence.

Zhipeng Liu

There are several general topics I would like to see discussed at the workshop:

  1. Different models in the KPZ universality class and recent progress on these models.
  2. Different approaches people have developed recently or are developing to study these models.
  3. The limiting fields, the KPZ fixed point, and the directed landscape—their properties and recent progress.
  4. Other potential directions that people are interested in.

Matteo Mucciconi

Here are a few questions:

  1. The probability law of large classes of natural Gibbsian line ensembles (or even Gibbsian surface/hyper-surface ensembles) can be proven to be logarithmically concave. Can this be used to show tightness in some interesting scaling limit?
  2. Related to (1): Are there higher-dimensional analogs of line ensembles that can be sampled in a reasonable way?
  3. What happened to the Inverse Beta polymers by Le Doussal–Thierry? Too difficult? Are they part of a line ensemble?
  4. Can TASEP in half-space be solved through some suitable variant of the RSK?

Tomohiro Sasamoto

I would be interested in discussing the following topics:

I may expand the description later.

Xiao Shen

I have been working on applying probabilistic methods to the study of KPZ universality, such as percolation and coupling. In this workshop, I am particularly interested in learning the probabilistic properties of the RSK and geometric RSK in these models and seeing whether these properties can be established directly using “corner flipping” induction in the analysis of increment-stationary KPZ models.

Evan Sorensen

I am interested in the structure of jointly invariant measures (or multi-type/colored invariant measures) for models in the KPZ universality class. Within the context of KPZ universality, these structures have been used to describe geometric features of infinite geodesics in KPZ models. For example:

More recently, jointly invariant measures have been used to describe the asymptotics of shocks in the KPZ equation (arXiv:2406.06502) and the covariance function of a limiting Gaussian process for the periodic KPZ equation (arXiv:2409.03613). Dauvergne and Virág (arXiv:2405.00194) recently gave an alternate construction of the directed landscape from Brownian motion using variants of the RSK correspondence. This opens the possibility of showing convergence to the directed landscape by using information about the joint distribution of jointly invariant measures. I am interested in seeing if these techniques can be extended to positive temperature models, such as the O’Connell–Yor polymer and the inverse-gamma polymer.

Mikhail Tikhonov

I would like to discuss several topics during the workshop:

  1. Recently, I was working with vertex models and I am interested in discussing KPZ universality and the stochastic six-vertex model with different boundary conditions; possibly expanding recent KPZ fixed point results from arXiv:2412.18117.
  2. I’m interested in KPZ behavior in quantum spin systems; especially in connections beyond Bethe Ansatz.
  3. I would like to understand how local transformations, like Yang–Baxter equations or analogues in particle systems, interplay with KPZ universality.

Li-Cheng Tsai

My research interests in this workshop include the following:

  1. Stochastic Heat Flow (SHF). The SHF is the scaling limit of 2+1 dimensional directed polymers around the critical temperature. The SHF can be viewed as the analog of the Stochastic Heat Equation (SHE) in 2D, though the former is much more intricate than the latter. To name a few recent results, the existence and axiomatic characterization of the SHF have been obtained, and the polymer measure of the SHF has been constructed. It will be interesting to study further properties of the SHF and its polymer measures.
  2. Formulas of the transition probabilities. This is a direction that I don’t know very much about but am interested in. Matetski, Quastel, and Remenik obtained an explicit formula of (or related to) the transition probabilities of TASEP and took the scaling limit to construct the KPZ fixed point. I’d like to see if we can find more examples of such formulas (beyond those already obtained) and if we can find a more conceptually direct derivation.

Hieu Trung Pham Vu

My interest in the workshop comes from the paper Anisotropic growth of random surfaces in 2+1 dimensions by Borodin and Ferrari. In this paper, they constructed a class of (2+1)-dimensional interacting particle systems, some of which have the property that their fixed-time projections can be viewed as a random tiling model, while certain (1+1)-dimensional projections represent models in the Kardar–Parisi–Zhang universality class of stochastic growth models. In 2024, we found the asymptotic curves separating various phases of a family of dimer models. The approach used was also from a system of recurrence relations that can be understood as a 2+1-dimensional discrete evolution equation. Our approach, however, has not shown any direct connection with the point of view of interacting particle systems and KPZ. My goal is to find an interacting particle system corresponding to this dimer model and confirm whether their (1+1)-projection represents any model in the KPZ universality class.

Jiyue Zeng

Proposing a Problem

Suppose we have a two-layer stationary geometric LPP model.

We are interested in the stationary version of this model where the weights are specified as follows:

\[\omega_{i, j} = \begin{cases} \mathrm{Geo}\left( rs \right), & i=j=2,\\[6pt] \mathrm{Geo}\left( r\sqrt{q} \right), & i=j> 2,\\[6pt] \mathrm{Geo}\left( \dfrac{\sqrt{q}}{s} \right), & j=1,\; i> 2,\\[6pt] \mathrm{Geo}\left( s\sqrt{q} \right), & j=2,\; i> 3,\\[6pt] 0, & \textrm{if } i=j=1,\\[6pt] 0, & \textrm{if } i=2,\; j=1,\\[6pt] \mathrm{Geo}(q), & \textrm{otherwise}. \end{cases}\]

Here, \(s \in (\sqrt{q}, 1)\), \(r \in (0,1/s)\), and \(q \in (0,1)\) is a fixed parameter. Let \(G_{N,N}\) denote the last passage percolation from \((1,1)\) to \((N,N)\).

We have found a formula for the distribution of \(G_{N,N}\) by using a shift argument and analytic continuation. We took the critical scaling and found the formula for the limiting distribution. We have the following phase diagrams. These results will be released possibly in a month.

Now, we would like to find the diagonal distribution if we scale \(s\) and \(r\) differently. We want to discover, for example, some Gaussian fluctuation if we do not scale \(s\) and \(r\) to \(1\). We want to find a complete phase diagram.

Lingfu Zhang

A major direction of my research is within the KPZ universality class and integrable probability, with focuses including:

  1. Constructing new universal objects, primarily through exact formulas.
  2. Establishing universality—i.e., showing that a wide range of stochastic processes, particularly those without exact-solvable structures, converge to the same universal objects.
  3. Proving meaningful probabilistic results for these models, such as geometric behaviors, mixing of interacting particle systems, and spectra of random operators, based on an understanding of their structures derived from universality.

Some of my recent work includes:

In this workshop, I am interested in learning about each of the three directions mentioned above. Some broad directions I am currently thinking about include:

Zhengye Zhou

I am interested in ASEP on a ring. More specifically, how to describe its full space-time fluctuations; any way to construct useful observables based on its algebraic structure; and whether there are universal KPZ features that can be explicitly derived.