KPZ universality of stationary current correlations: the scaling form of the density correlator $S(j,t)$ and the $-|t|^{-4/3}$ decay of the current correlator $h(t)$ for non-reversible one-dimensional lattice gases

Ferrari and Spohn (SIGMA 12 (2016) 074) analyzed the two-time covariance of the height of one-dimensional KPZ growth and, for stationary driven lattice gases, derived from a sum rule linking current fluctuations to the stationary density correlator $S(j,t)$ that the current correlator should satisfy $h(t)\simeq -c_0|t|^{-4/3}$ and that the time-integrated current should have the covariance of fractional Brownian motion with Hurst exponent $1/3$ — while stating explicitly that the KPZ scaling form of $S(j,t)$ was proved only for TASEP, and that no spectral, sign, or monotonicity theorem supports $h$ in the non-reversible case. The model-independent core of the current side: prove, for non-reversible KPZ-class lattice gases — the canonical benchmark being ASEP with $p\neq q$ at density $\rho=1/2$, where the characteristic velocity vanishes — that the stationary density correlator has the universal KPZ scaling form with the same scaling function as stationary TASEP, that $h(t)<0$ is monotone and decays as $-|t|^{-4/3}$ with the predicted prefactor, and that the characteristic-integrated current has the Hurst-$1/3$ covariance; or refute the predicted structure by a rigorous counterexample within the class.

Background

Interfaces growing on a one-dimensional substrate in the Kardar–Parisi–Zhang (KPZ) universality class — the KPZ stochastic PDE itself, the totally asymmetric simple exclusion process (TASEP), polynuclear growth, and Eden-type growth — have height fluctuations of order $t^{1/3}$ and spatial correlations of order $t^{2/3}$ at large time $t$. TASEP is a lattice gas on $\mathbb{Z}$ with at most one particle per site, particles jumping right at rate $1$ into empty sites; its hydrodynamic current is $j(\rho)=\rho(1-\rho)$. Via a height function built from the time-integrated current across a lattice bond together with the instantaneous particle configuration, TASEP maps to last-passage percolation (LPP), the model of directed lattice paths that maximize accumulated random passage times, whose scaling limits are the Airy processes: the Airy$_2$ process for droplet (step) initial data, Airy$_1$ for flat data, and Airy$_\text{stat}$ for stationary data, with one-point marginals given by the Tracy–Widom distributions of random-matrix theory (GUE and GOE, the Gaussian unitary and orthogonal ensembles) and, in the stationary case, the Baik–Rains distribution $F_\text{BR}$; the Airy processes are locally Brownian on the $t^{2/3}$ scale. Ferrari and Spohn (SIGMA 12 (2016) 074) considered the most basic temporal observable, the two-time height covariance $C(t_0,t)=\mathrm{Cov}(h(0,t_0),h(0,t))$ in the large-$t$ limit with $\tau=t_0/t$ fixed. Combining slow decorrelation (fluctuations of order $t^{1/3}$ persist along characteristics of the hydrodynamic equation) with a decomposition of LPP paths, they expressed the rescaled pair through a variational problem involving two independent Airy-type processes, e.g. for step data $\mathcal{R}^\mathrm{step}(1)=\tau^{1/3}\max_{u\in\mathbb{R}}\{\mathcal{A}_2(u)-u^2+\hat\tau^{-1/3}\tilde{\mathcal{A}}_2(u\hat\tau^{2/3})-u^2\hat\tau\}$ with $\hat\tau=\tau/(1-\tau)$, and concluded $C^\mathrm{step}(\tau)=\mathcal{O}(\tau^{2/3})$ and $C^\mathrm{flat}(\tau)=\mathcal{O}(\tau^{4/3})$ as $\tau\to0$, $C^\mathrm{step}(\tau)=\mathrm{Var}(\xi_\mathrm{GUE})-\tfrac12\mathrm{Var}(\xi_\mathrm{BR})(1-\tau)^{2/3}+\mathcal{O}(1-\tau)$ as $\tau\to1$ (similarly for flat, with $2^{-4/3}\mathrm{Var}(\xi_\mathrm{GOE})$), and, for stationary data, the exact formula $C^\mathrm{stat}(\tau)=\tfrac12\mathrm{Var}(\xi_\mathrm{BR})(1+\tau^{2/3}-(1-\tau)^{2/3})$. These derivations use the locally Brownian character of the Airy processes and were stated by the authors not to be mathematically rigorous; the announced Corwin–Hammond work was a private communication and never appeared as such. The same paper then connected time correlations to transport: for a general one-dimensional stochastic lattice gas with exchange rates $c_{j,j+1}(\eta)>0$, stationary measure $\mu_\rho$ at density $\rho$, and macroscopic current $j(\rho)$, the height at the origin is the time-integrated current $J(t)$, whose increments have covariance $\mathrm{Cov}(\mathsf{j}(t),\mathsf{j}(t'))=\langle c_{0,1}\rangle_\rho\delta(t-t')+h(t-t')$ with $h(t)=-\langle(r^\mathrm{R}_{0,1}-j(\rho))e^{L|t|}(r_{0,1}-j(\rho))\rangle_\rho$ ($r$ the microscopic current function, $r^\mathrm{R}$ the time-reversed current function, and $L$ the generator of the dynamics). Particle conservation yields the sum rule $\mathrm{Var}(J(t))=\sum_j|j|S(j,t)-\sum_j|j|S(j,0)$, linking current fluctuations to the stationary density correlation $S(j,t)=\mathrm{Cov}(\eta_j(t),\eta_0(0))$, with susceptibility $\chi=\sum_jS(j,0)$. For reversible models the generator is self-adjoint in $L^2(\mu_\rho)$, so $h(t)=-\int_0^\infty e^{-\lambda|t|}\nu(d\lambda)$ is negative and increasing, and diffusive broadening $S(j,t)\simeq\chi(Dt)^{-1/2}f_\mathrm{G}((Dt)^{-1/2}j)$ forces $h(t)\simeq-t^{-3/2}$, so $J(t)$ has fractional-Brownian covariance with Hurst exponent $1/4$. For non-reversible KPZ-class models, KPZ scaling theory instead predicts $S(j,t)\simeq\chi(\Gamma t)^{-2/3}f_\mathrm{KPZ}((\Gamma t)^{-2/3}(j-v(\rho)t))$ with $\Gamma=\tfrac12\chi^2|j''(\rho)|$, from which the sum rule gives $h(t)\simeq-c_0|t|^{-4/3}$ with $c_0=\tfrac19\Gamma^{2/3}\chi\int_\mathbb{R}|x|f_\mathrm{KPZ}(x)dx$, and $\mathrm{Cov}(J(\tau t),J(t))\simeq\tfrac12(1+\tau^{2/3}-(1-\tau)^{2/3})(\Gamma t)^{2/3}\chi\int_\mathbb{R}|x|f_\mathrm{KPZ}(x)dx$ — the covariance of fractional Brownian motion with Hurst exponent $1/3$, although the process itself is non-Gaussian with Baik–Rains marginals. Monte Carlo simulations of stationary TASEP confirmed $h<0$, monotonicity, and the $-t^{-4/3}$ law. But the scaling form of $S(j,t)$ was proved only for stationary TASEP, via the determinantal analysis of Ferrari–Spohn (Comm. Math. Phys. 265 (2006) 1–44), where determinantal means that correlation functions reduce to determinants of an explicit kernel, and in which the scaling function was extracted from the transition probabilities of a second-class particle (a marked particle that exchanges positions with ordinary ones) and identified within the Baik–Rains family; and for non-reversible dynamics no analogue of the self-adjoint spectral formula was available, so even the sign and monotonicity of $h$ rested on assumption. A decade of subsequent work made the covariance asymptotics and explicit multi-time joint laws rigorous — but exclusively for models with integrable (LPP/polymer/PNG or KPZ-fixed-point) structure. The sector on which this problem record focuses, the model-independent current-correlation structure for genuinely non-reversible KPZ-class lattice gases, has remained conjectural: no rigorous $S(j,t)$ or $h(t)$ result exists for the asymmetric simple exclusion process (ASEP, the nearest-neighbor exclusion process with right and left jump rates $p\neq q$), and the 2025 state of the art (Ferrari–Liu, J. Stat. Phys. 192 (2025)) treats universality for ASEP and speed-changed exclusion processes as conjectures supported by numerics.

Problem Statement

Consider a one-dimensional stochastic lattice gas on $\mathbb{Z}$ with translation-invariant finite-range exchange rates $c_{j,j+1}(\eta)>0$ and at most one particle per site, admitting a unique stationary measure $\mu_\rho$ at each density $\rho$ with exponential space-mixing, with concave hydrodynamic current $j(\rho)$, and in the KPZ universality class (i.e. $j''(\rho)\neq0$); the dynamics is assumed non-reversible with respect to $\mu_\rho$, meaning the microscopic process has a non-vanishing stationary current. The canonical benchmarks are the asymmetric simple exclusion process (ASEP) with right/left jump rates $p\neq q$, $p+q=1$, at $\rho=1/2$ where the characteristic velocity $v(\rho)=j'(\rho)$ vanishes (interpolating between TASEP at $q=0$ and the reversible diffusive limit $p=q$), and the speed-changed exclusion processes of Ferrari–Liu (2025), which have exactly known stationary measures but no determinantal structure. Write $S(j,t)=\mathrm{Cov}(\eta_j(t),\eta_0(0))$ for the stationary density correlation, $\chi=\sum_jS(j,0)$ for the susceptibility, $\Gamma=\tfrac12\chi^2|j''(\rho)|$ for the KPZ amplitude constant, and $\mathrm{Cov}(\mathsf{j}(t),\mathsf{j}(t'))=\langle c_{0,1}\rangle_\rho\delta(t-t')+h(t-t')$ for the stationary current covariance with smooth part $h(t)=-\langle(r^\mathrm{R}_{0,1}-j(\rho))e^{L|t|}(r_{0,1}-j(\rho))\rangle_\rho$. Establish, at theorem level, for non-reversible KPZ-class lattice gases of this kind — for which the scaling form in (a) is currently proved only for stationary TASEP, while the sign, monotonicity, and decay of $h$ in (b) and the spectral/representation structure in (c) remain open even for TASEP itself: (a) the KPZ scaling form of the stationary density correlator $S(j,t)\simeq\chi(\Gamma t)^{-2/3}f_\mathrm{KPZ}((\Gamma t)^{-2/3}(j-v(\rho)t))$ as $t\to\infty$, with a universal scaling function $f_\mathrm{KPZ}$ that, after this rescaling, coincides with the stationary-TASEP scaling function of Ferrari–Spohn (2006) (equivalently, with the Baik–Rains distribution family governing the stationary KPZ fixed point); (b) the sign, monotonicity, and decay of the current correlator, namely $h(t)<0$ and non-decreasing in $|t|$ with $h(t)\to0$, and $h(t)\simeq-c_0|t|^{-4/3}$ with $c_0=\tfrac19\Gamma^{2/3}\chi\int_\mathbb{R}|x|f_\mathrm{KPZ}(x)dx$; and consequently the covariance of the current $J(t)$ integrated along the characteristic ray $\{x=v(\rho)t\}$, $\mathrm{Cov}(J(\tau t),J(t))\simeq\tfrac12(1+\tau^{2/3}-(1-\tau)^{2/3})(\Gamma t)^{2/3}\chi\int_\mathbb{R}|x|f_\mathrm{KPZ}(x)dx$, i.e. the fractional-Brownian covariance with Hurst exponent $1/3$ (with non-Gaussian one-point Baik–Rains-type marginals); and (c), optionally, a structural strengthening that is not required for resolution — a theorem-level foundation for these statements for the non-reversible generator, for instance a spectral, sign, or representation theorem for $h(t)$ replacing the self-adjoint reversible formula $h(t)=-\int_0^\infty e^{-\lambda|t|}\nu(d\lambda)$. The problem is answered by a rigorous derivation of (a) and (b) — including the covariance of the characteristic-integrated current — valid either for the general class of non-reversible KPZ-class lattice gases defined above or for an explicitly named sub-class that strictly extends the already-solved stationary-TASEP/integrable cases (ASEP with $p\neq q$ at $\rho=1/2$ being the canonical such benchmark); or refuted by a rigorous counterexample within the class violating the predicted sign or monotonicity of $h$, the exponents, the prefactor, the Hurst-$1/3$ covariance form, or the claimed coincidence of the $S(j,t)$ scaling function with the stationary-TASEP (Baik–Rains) function.

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