Symmetric elliptic functions, IRF models, and dynamic exclusion processes

Abstract

We introduce stochastic Interaction-Round-a-Face (IRF) models that are related to representations of the elliptic quantum group E τ,η (sl 2 ). For stochasic IRF models in a quadrant, we evaluate averages for a broad family of observables that can be viewed as higher analogs of q-moments of the height function for the stochastic (higher spin) six vertex models.In a certain limit, the stochastic IRF models degenerate to (1+1)d interacting particle systems that we call dynamic ASEP and SSEP; their jump rates depend on local values of the height function. For the step initial condition, we evaluate averages of observables for them as well, and use those to investigate one-point asymptotics of the dynamic SSEP.The construction and proofs are based on remarkable properties (branching and Pieri rules, Cauchy identities) of a (seemingly new) family of symmetric elliptic functions that arise as matrix elements in an infinite volume limit of the algebraic Bethe ansatz for E τ,η (sl 2 ).SYMMETRIC ELLIPTIC FUNCTIONS, IRF MODELS, AND DYNAMIC EXCLUSION PROCESSES 2 foundational works. This article is concerned with two of its very recent and closely related applications to:(a) theory of symmetric functions, with new families of symmetric functions being introduced and new summation identities for older ones being proved, see [7,35,31,32,15,16,26];(b) deriving new exact formulas for averages of observables in two-dimensional integrable lattice models and (1+1)-dimensional random growth models, and using those to study large scale and time asymptotics [18,15,16,4,1,8,14,9,22].All the above cited papers deal with the R-matrix for the (higher spin) six vertex model and its degenerations (in other words, with representations of the affine quantum group U q (ŝl 2 ) and their limits). Certain progress has been achieved for the U q (ŝl n ) case as well, with new Markov chains introduced via the corresponding R-matrices [30,17] and a duality functional for them provided in [29].The goal of this work is to climb higher in the hierarchy and to extend some of the recent progress from vertex models to the so-called Interaction-Round-a-Face (IRF) models, also known as face and solid-on-solid (SOS) models. The corresponding R-matrices satisfy a face version of the star-triangle relation also known as the dynamical Yang-Baxter equation.The IRF models were originally introduced by Baxter [5] as a tool to analyze the eight vertex model, but they quickly became a subject on their own, see, e. g., [6,36] and references therein. We will only be concerned with the sl 2 case, the basic instance of which is due to the original work [5] and is often called the eight vertex SOS model, and whose fused versions were introduced and extensively studied in [21,19,20].The IRF models were framed in a representation theoretic language by Felder [23], who introduced the concept of an elliptic quantum group. The simplest instance is the quantum elliptic group E τ,η (sl 2 ); the corresponding representation theory and the algebraic Bethe ansatz were developed by F...

Stochastic PDE Limit of the Six Vertex Model

“…The next development regarding dynamic ASEP was the Markov duality derived by [BC17] between it and the standard ASEP (Section 2 herein). The duality function which intertwines these two processes is the same observable which had arisen in the earlier work of [Bor17]. [BC17] also derived a translation invariant, stationary measure for the dynamic ASEP (see Definition 1.2).…”

“…Before going into greater depth about our present contribution, let us recall the previous work on this process. Besides introducing the model, [Bor17] developed a generalization of the method introduced by [BP18, BP16] (in studying non-dynamic higher spin vertex models) in order to compute contour integral formulas for expectations of a class of observables for certain initial conditions (in particular for the wedge, where s 0 (x) = |x|). Taking the limit q → 1 leads to a dynamic version of the SSEP.…”

“…Taking the limit q → 1 leads to a dynamic version of the SSEP. In that limit, the observables and formulas simplify sufficiently so that [Bor17] was able to probe some asymptotics (for wedge initial data).…”

“…[Agg18] introduced dynamic analogs of the higher spin vertex models from [CP16], and [BM18] and [ABB18] introduced other types of dynamic analogs of growth models and vertex models (i.e., the growth rates depend on the height through an additional dynamic parameter). For the dynamic stochastic six vertex model, [BG18] used the formulas from [Bor17] to derive a law of large numbers and Gaussian central limit theorem under a particular choice of scaling.…”

Stochasticization of Solutions to the Yang–Baxter Equation

“…Based on the framework of elliptic quantum groups [29,30,31], a family of stochastic dynamical vertex, or interaction round-a-face (IRF), models were proposed in [14], which were later generalized in [1,42] and analyzed from a probabilistic perspective in [16]. However, in order to ensure stochasticity of these models, the works [1,14,42] took trigonometric degenerations of the originally elliptic solutions to the dynamical Yang-Baxter equation. Thus, the ellipticity of the stochastic vertex weights was lost.…”