The stochastic eight-vertex (S8V) model introduced by Funaki, Nishijima and Suda is a discrete-time interacting particle system on the one-dimensional integer lattice, whose space-time updates are governed by a stochastic vertex matrix that randomly keeps, permutes, creates or annihilates particle pairs, so that the total particle number is not conserved. Under a balance condition on the vertex weights, the Bernoulli product measures are invariant, and the authors formally derived the hydrodynamic limit, the linear fluctuation limit, and a nonlinear limit governed by a new type of KPZ–Burgers stochastic partial differential equation (SPDE) containing a Burgers-type nonlinearity and two independent space–time white noises, one entering in divergence (spatial-derivative) form. All fluctuation-limit arguments were explicitly formal: they assume unproved first- and second-order Boltzmann–Gibbs replacement principles, and the well-posedness of the limiting SPDE and the classification of the model's invariant measures were deferred to future work. This problem asks for the rigorous completion of that program: prove the Boltzmann–Gibbs replacement principles for the nonconservative S8V dynamics, prove convergence of the weakly asymmetric fluctuation field to the two-noise KPZ–Burgers SPDE, establish well-posedness of that SPDE together with a characterization of its invariant measures, and classify all translation-invariant invariant measures of the bi-infinite S8V beyond the balanced Bernoulli products. No resolution of any of these items is known in the literature to date.

Background

The eight-vertex model of statistical mechanics assigns to each vertex of the square lattice one of eight arrow configurations, weighted so that each vertex has an even number of arrows pointing in: six configurations obey the ice rule (two arrows in, two arrows out), and the remaining two act as a source (four arrows out) or a sink (four arrows in). Funaki, Nishijima and Suda introduced a stochastic counterpart, the stochastic eight-vertex (S8V) model: a discrete-time Markov chain on particle configurations ηt∈{0,1}Z\eta_t\in\{0,1\}^{\mathbb{Z}} of the one-dimensional integer lattice, naturally viewed as the occupation of the horizontal edges of a two-dimensional space-time lattice whose vertical direction is discrete time. Each vertex of the space-time lattice receives the two particle variables on its left and bottom edges and emits two on its right and top edges according to a stochastic vertex matrix LL: if the two inputs are equal, the vertex either leaves the pair unchanged or creates/annihilates it (state (0,0)(0,0) is sent to (0,0)(0,0) with probability b3b_3 and to (1,1)(1,1) with probability 1−b31-b_3; state (1,1)(1,1) is sent to (1,1)(1,1) with probability b4b_4 and to (0,0)(0,0) with probability 1−b41-b_4); if the two inputs differ, the vertex keeps or swaps them (with probabilities b1b_1, 1−b11-b_1 and b2b_2, 1−b21-b_2). Because the source and sink moves change the particle number, the dynamics is genuinely nonconservative, and the bi-infinite (indexed by all of Z\mathbb{Z}) version of the process is constructed by an explicit path-dependent recursion. This should be contrasted with the stochastic six-vertex model and its higher-spin generalizations, which are conservative and possess rich integrable structure — the Yang–Baxter equation (a consistency relation on vertex weights that makes the model exactly solvable), Markov duality (an intertwining of two Markov processes that renders certain nonlocal observables computable), and Macdonald-measure machinery; those tools have been used to prove, for example, that the six-vertex model's height fluctuation converges to the KPZ equation and to identify the stochastic PDE limits of the stochastic six-vertex model. Funaki, Nishijima and Suda introduced their S8V model precisely because integrable methods fail for it: creation and annihilation of particles break the conservation law on which duality arguments rest, and the authors state that more robust probabilistic methods need to be developed. The KPZ equation (named after Kardar, Parisi and Zhang) is the singular SPDE ∂th=12∂x2h+12(∂xh)2+ξ\partial_t h=\tfrac12\partial_x^2h+\tfrac12(\partial_xh)^2+\xi with ξ\xi space–time white noise; it is the universal scaling limit of weakly asymmetric conservative one-dimensional systems, and the Burgers equation obtained by differentiating in space is its density-level equivalent. The hydrodynamic limit is the deterministic macroscopic density evolution that emerges from a microscopic particle system under diffusive space–time rescaling. A rigorous route from a microscopic system to KPZ/Burgers fluctuation limits goes through the Boltzmann–Gibbs principle: in its first-order form it states that a long but microscopic average of a local function of the configuration is asymptotically replaced by that function evaluated at the local empirical density (local equilibrium); a second-order refinement keeps the leading fluctuation correction, and it is this second-order term that produces the quadratic nonlinearity in the KPZ/Burgers limit. For conservative dynamics such principles are classical, but for a nonconservative process with creation and annihilation they have not been established. In the S8V paper the authors prove that the Bernoulli product measure with density ρ\rho (independent site occupations) is invariant for the bi-infinite process exactly under the balance condition (1−ρ)2(1−b2)(1−b3)=ρ2(1−b1)(1−b4)(1-\rho)^2(1-b_2)(1-b_3)=\rho^2(1-b_1)(1-b_4) on the vertex weights (their equation (2.15)), and they derive formally — under diffusive scaling, weak asymmetry b2=b1e−cεb_2=b_1e^{-c\sqrt\varepsilon} of order ε\sqrt\varepsilon, and pair creation/annihilation of order ε2\varepsilon^2 (b3=1−ε2c3b_3=1-\varepsilon^2c_3, b4=1−ε2c4b_4=1-\varepsilon^2c_4) — a nonlinear fluctuation limit governed by an SPDE of the form ∂tY=v∗2∂u2Y−κ∗2∂u(Y2)−λ∗2Y+D1 ∂uW˙1+D2 W˙2\partial_tY=\frac{v_*}{2}\partial_u^2Y-\frac{\kappa_*}{2}\partial_u(Y^2)-\frac{\lambda_*}{2}Y+\sqrt{D_1}\,\partial_u\dot W_1+\sqrt{D_2}\,\dot W_2, where W˙1,W˙2\dot W_1,\dot W_2 are independent space–time white noises and the coefficients are explicit functions of the microscopic vertex weights, reported in the source as v∗=2b11−b1v_*=\frac{2b_1}{1-b_1}, κ∗=2b1c1−b1\kappa_*=\frac{2b_1c}{1-b_1}, λ∗=8(c3(1−ρ)+c4ρ)\lambda_*=8(c_3(1-\rho)+c_4\rho), D1=2b1ρ(1−ρ)1−b1D_1=\frac{2b_1\rho(1-\rho)}{1-b_1}, D2=8ρ2c4D_2=8\rho^2c_4: a KPZ–Burgers equation with an additional independent additive noise. The novelty and the difficulty both lie in the two-noise structure (the conservative noise entering as a spatial derivative and the nonconservative noise entering additively) and in the absence of integrable structure. The paper states that a rigorous derivation, the analytic well-posedness (existence, uniqueness, and any needed renormalization) of this SPDE together with its invariant measures, the proofs of the Boltzmann–Gibbs principles for the nonconservative dynamics, and the classification of all translation-invariant invariant measures of the bi-infinite S8V (beyond the balanced Bernoulli products, deferred in their Remark 2.2-(2)) remain open.

Problem Statement

For the stochastic eight-vertex (S8V) model of Funaki–Nishijima–Suda — the discrete-time Markov chain on particle configurations {0,1}Z\{0,1\}^{\mathbb{Z}} whose space-time vertex update matrix carries the weights b1,…,b4b_1,\dots,b_4 as defined above — establish a rigorous fluctuation-limit theory in the weakly asymmetric regime in which the conservative asymmetry is scaled at order ε\sqrt\varepsilon (b2=b1e−cεb_2=b_1e^{-c\sqrt\varepsilon}) and the pair-creation/annihilation probabilities at order ε2\varepsilon^2 (b3=1−ε2c3b_3=1-\varepsilon^2c_3, b4=1−ε2c4b_4=1-\varepsilon^2c_4), consisting of: (1) a proof of the first-order and second-order Boltzmann–Gibbs replacement principles for the nonconservative S8V dynamics (one-block and two-block estimates for local empirical averages under local equilibrium, without integrable structure); (2) a proof of convergence of the microscopic density fluctuation field, in the sense of martingale problems or of finite-dimensional distributions, in the moving frame of the characteristic speed of the macroscopic equation, to the unique solution of the nonlinear KPZ–Burgers SPDE obtained formally in the source, i.e. an equation of the form ∂tY=v∗2∂u2Y−κ∗2∂u(Y2)−λ∗2Y+D1∂uW˙1+D2W˙2\partial_tY=\frac{v_*}{2}\partial_u^2Y-\frac{\kappa_*}{2}\partial_u(Y^2)-\frac{\lambda_*}{2}Y+\sqrt{D_1}\partial_u\dot W_1+\sqrt{D_2}\dot W_2, the SPDE with Burgers-type nonlinearity, damping, and two independent space–time white noises of which one enters in divergence form and the other additively, the explicit coefficients (functions of the microscopic vertex weights and of the scaling parameters) and the characteristic speed being those derived in the source's formal scaling analysis; (3) a well-posedness theory for that SPDE — existence, uniqueness (up to any required renormalization), and a characterization of its invariant measures; and (4) for the bi-infinite S8V model with vertex weights satisfying the balance condition (1−ρ)2(1−b2)(1−b3)=ρ2(1−b1)(1−b4)(1-\rho)^2(1-b_2)(1-b_3)=\rho^2(1-b_1)(1-b_4) (equation (2.15) of the source), a classification of all its translation-invariant invariant measures beyond the Bernoulli product measures. Each of these items is explicitly deferred in the source: the fluctuation-limit arguments assume the unproved Boltzmann–Gibbs replacement principles, the well-posedness of the two-noise limiting SPDE and the description of its invariant measures are announced as the subject of a separate paper, and the classification of translation-invariant invariant measures of the bi-infinite S8V beyond the balanced Bernoulli products is left open. The authors' methodological remark that robust probabilistic methods should replace unavailable integrable-probability techniques is motivation for this program, not a separate deliverable. Items (1)–(3) are cumulative component steps of one program, the fluctuation-limit theory in the weakly asymmetric regime — with the replacement principles (1) the intermediate estimates consumed by the convergence argument (2), so that proving (2) discharges (1), and the well-posedness theory (3) supplying the existence and uniqueness that (2) invokes — whereas item (4) is independent of this scaling analysis and concerns the equilibrium of the bi-infinite S8V for general weights satisfying the balance condition.

References

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