Bulk Gaussian (Edwards–Wilkinson) scaling limits of the $(2+1)$-dimensional $q$-Whittaker growth model at fixed $q\in(0,1)$

The $q$-Whittaker particle system, a push-block dynamics on interlacing particle arrays whose marginals are $q$-Whittaker processes, defines an exactly solvable $(2+1)$-dimensional anisotropic interface growth model. At $q=0$ (Schur processes) its height function has Gaussian-free-field statistics at a single long time, and Borodin–Corwin–Ferrari proved an Edwards–Wilkinson-type space-time limit in the $q\to1$ regime by passing through Gaussian (Whittaker) stochastic differential equation limits with intermediate large-time and large-system limits; they state explicitly that they do not know how to prove an analogous set of results for the $q$-Whittaker process at fixed $q\in(0,1)$. Later work of Yu-Ting Chen strengthened the $q\to1$ chain to process-level convergence to the additive stochastic heat equation and extended the rescaled time range to the whole half-line via time inversion, but only for the Gaussian limiting SDE systems, not for the particle system at fixed $q$. The open problem is to prove the analogous bulk scaling limit directly at fixed $q\in(0,1)$: the macroscopic law of large numbers, convergence of diffusively rescaled height fluctuations to the two-dimensional log-correlated Gaussian free field, and the Edwards–Wilkinson space-time evolution along characteristics for all rescaled times — or else to rigorously identify a different universality class.

Background

A $(2+1)$-dimensional stochastic growth model is a Markov process on integer-valued height functions $h(x,t)$, $x\in\mathbb Z^2$, $t\ge0$, describing an interface that grows by random local deposition and rearrangement. Two universality classes govern the large-scale fluctuations of such interfaces. In the Kardar–Parisi–Zhang (KPZ) class the lateral-growth nonlinearity is relevant and fluctuations are non-Gaussian; in the Edwards–Wilkinson (EW) class the nonlinearity is irrelevant and the centered height field is expected to converge to the solution of the additive stochastic heat equation $\partial_t u=\nu\Delta u+\sigma\dot W$, where $\dot W$ is space-time white noise; this equation is also called the EW equation, and its fixed-time statistics are those of the two-dimensional Gaussian free field (GFF), the log-correlated generalized Gaussian field that is the canonical Gaussian random surface in two dimensions. Wolf (1991) predicted that the class is decided by the sign of the determinant of the Hessian of the mean growth velocity viewed as a function of the local slope: a negative determinant (the anisotropic KPZ, or AKPZ, signature, arising when growth along the two coordinate directions competes rather than cooperates) forces EW behavior, with possible logarithmic corrections; otherwise KPZ behavior. Prähofer–Spohn (1997) gave the first exactly solved model in the anisotropic regime.

The model under study is the $q$-Whittaker particle system, a continuous-time interacting particle system on interlacing arrays (Gelfand–Tsetlin patterns): triangular arrays of nonnegative integers $\lambda^{(n)}_k$, $1\le k\le n\le N$, whose adjacent levels interlace. Each particle carries an exponential clock whose rate depends, through powers of a parameter $q\in(0,1)$, on the gaps to its left and upper neighbors; when a clock rings, the particle moves right by one and pushes the entire vertical string of particles above it that share its position (the push-block mechanism). These dynamics preserve the class of $q$-Whittaker processes, probability measures on interlacing arrays built from the $q$-Whittaker symmetric functions (the $t=0$ degeneration of Macdonald polynomials), and were introduced by Borodin–Corwin within the Macdonald-process framework; exact contour-integral formulas for the moments of many observables follow from eigenrelations for $q$-Whittaker functions, as recorded by Borodin–Corwin–Gorin–Shakirov. Standard choices of specializations include the Plancherel one (a single parameter $\gamma$) and the alpha one (inhomogeneous weights). Via a lozenge-tiling encoding, configurations define a growing $(2+1)$-dimensional height function; the pushing mechanism induces a preferred tilt direction, so the model is believed to lie in the anisotropic KPZ class with the AKPZ signature, i.e. in Wolf's EW scenario.

The two endpoints of the $q$-range are understood. At $q=0$ the dynamics degenerate to Schur-process growth, whose determinantal structure allowed Borodin–Ferrari to prove that the interface fluctuates on the scale of $\ln\gamma$ (logarithmic in time) and that, after centering and diffusive rescaling, it has GFF statistics at a single long time. At the other endpoint, Borodin–Corwin–Ferrari (BCF) study the $q\to1$ regime: setting $q=e^{-\varepsilon}$, they derive law-of-large-numbers ODEs with explicit integral solutions, central-limit-theorem fluctuation SDEs, and, after a further large-time diffusive limit to a Gaussian SDE system $\zeta$ and a steepest-descent analysis of exact covariance formulas in the limit of infinitely many levels $N\to\infty$, they prove that the space-time covariances of $\zeta$ along characteristic rays match those of the EW equation in equilibrium, so that the bulk height function converges to an evolving GFF. Two limitations are stated explicitly by BCF. First, the matching holds only for a rescaled time interval $t\in[0,1]$: the authors say they have no explanation for this time dilation, suspect it results from their intermediate limits, and note that Wolf's prediction suggests a direct $N\to\infty$ limit of the $q$-Whittaker particle system would produce the EW equation for all $t\in[0,\infty)$. Second, the whole chain passes through the $q\to1$ Gaussian SDE limit: BCF write that they do not know how to prove an analogous set of results for the $q$-Whittaker process except for certain Gaussian limits of it arising as $q\to1$, while it is reasonable to expect the same asymptotic behavior across the whole range $q\in(0,1)$. For the torus version of the system (introduced by Corwin–Toninelli, who identified its stationary measure), Borodin–Corwin–Toninelli proved the same $q\to1$ strategy yields Wolf's prediction at the level of covariances. Subsequently Yu-Ting Chen strengthened the Gaussian chain considerably: the rescaled Whittaker driven SDEs converge weakly, as distribution-valued processes, to the additive stochastic heat equation, and the $\zeta$ system converges to the time-inverted stationary additive stochastic heat equation over the entire rescaled time half-line, which answers, within the Gaussian approximation, the time-interval question raised by BCF. On the continuum side, Gu proved that the two-dimensional KPZ equation with a logarithmically tuned nonlinearity and small coupling scales to the EW equation with an effective variance, while Cannizzaro–Erhard–Schönbauer and Cannizzaro–Erhard–Toninelli constructed and analyzed the stationary anisotropic KPZ equation, proving it is logarithmically superdiffusive — logarithmic corrections to diffusive behavior consistent with Wolf's renormalization-group picture, and in contrast to the naive belief that it simply converges to the two-dimensional additive stochastic heat equation.

Despite these advances, nothing is known about the bulk large-scale statistics of the $q$-Whittaker growth model at fixed $q\in(0,1)$: the determinantal structure available at $q=0$ and the Gaussian (Whittaker-SDE) reduction available at $q\to1$ are both unavailable, and the exact fixed-$q$ moment formulas have never been asymptotically analyzed in the large-system, long-time limit. This is the problem recorded here.

Problem Statement

For fixed $q\in(0,1)$ — with $q$ not scaled to $1$ along with system size or time — prove the bulk large-scale Gaussian limit of the $(2+1)$-dimensional interface growth model generated by the $q$-Whittaker particle system (the push-block dynamics on interlacing arrays preserving $q$-Whittaker processes, with, for example, Plancherel or alpha specializations and packed initial data), directly, without passing through the $q\to1$ Gaussian (Whittaker-SDE) approximation. Concretely, establish the analog of the results known at $q=0$ and in the $q\to1$ regime:

1. a law of large numbers identifying the macroscopic shape and particle trajectories in the joint large-system, long-time limit (the fixed-$q$ analog of the source's LLN);
2. the centered, diffusively rescaled height field at fixed macroscopic time converges — at least at the level of covariances / finite-dimensional distributions — to the two-dimensional log-correlated Gaussian free field;
3. the space-time fluctuations along characteristic rays are governed, for the entire rescaled time range $t\in[0,\infty)$ (not only a bounded interval), by the $(2+1)$-dimensional Edwards–Wilkinson equation, i.e. the additive stochastic heat equation, possibly in its stationary regime with the time-inversion structure discovered in the Gaussian regime; process-level (tightness) convergence is the natural stronger form, beyond the required covariance-level matching.

If the Gaussian/Edwards–Wilkinson picture fails at fixed $q$, an admissible answer is a rigorous proof of the actual universality class of the bulk fluctuations (for example a limit with logarithmic corrections), together with the corresponding fixed-time and space-time limit theorems. The expected outcome, per the source ('it is reasonable to expect that the asymptotic behavior remains the same for the whole range of $q\in(0,1)$'), is the Gaussian-free-field / Edwards–Wilkinson limit, but a rigorously proved alternative identification is equally valid. No restriction on method is imposed: asymptotic analysis of the exact fixed-$q$ moment formulas, a fluctuation theory for the dynamics itself, or any other route. Results that establish only the fixed-time Gaussian-free-field covariance (the $q=0$-strength result) without the space-time evolution would constitute partial progress, since the source's analogous set of results includes the space-time Edwards–Wilkinson matching.

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