The infinite-dimensional limit of KPZ surface growth and the upper critical dimension of the KPZ universality class

Does the Kardar–Parisi–Zhang (KPZ) universality class of stochastic surface growth possess a finite upper critical dimension $d_u$, above which growing interfaces become asymptotically flat and behave as in the linear Edwards–Wilkinson (EW) class, or does the KPZ nonlinearity remain relevant at every finite dimension, with EW-like behavior emerging only in the $d\to\infty$ limit? This decades-old question was sharpened by Oliveira (Europhys. Lett. 133, 28001 (2021)), who showed that earlier Bethe-lattice growth simulations were dominated by Cayley-tree boundary effects and that, at the central site of finite trees (an approximation to the Bethe lattice), both EW- and KPZ-class models produce smooth, similar height fluctuations — behavior expected for $d\geq d_u$. The open problem left by that work is: decide the fate of the KPZ nonlinearity in high and infinite dimension; develop exact Bethe-lattice (mean-field) solutions for stochastic growth dynamics in the spirit of exact equilibrium solutions such as the Ising model on the Bethe lattice; place the Cayley-tree chemical-distance scaling and the central-site one-point diagnostics used to probe the infinite-dimensional limit on rigorous footing; and characterize nonrelaxational KPZ rules such as ballistic deposition on effectively infinite-dimensional substrates. The central question remains explicitly unresolved: analytic predictions remain split between $d_u<4$, $d_u>4$, and $d_u=\infty$, simulations up to $d=15$ show no crossover, and no exact solution method for growth models on the Bethe lattice exists, although 2026 preprints have clarified the complete-graph behavior of continuum KPZ (irrelevant nonlinearity at $d=\infty$) and of ballistic deposition (which loses universality there).

Background

A vast family of nonequilibrium interface growth processes — vapor deposition, bacterial colony fronts, driven diffusive systems, directed polymers in random media — is described at long scales by the KPZ stochastic partial differential equation $\partial_t h=\nu\nabla^2 h+(\lambda/2)(\nabla h)^2+\eta$, where $h(\mathbf{x},t)$ is the interface height, $\nu$ is a surface tension, $\lambda$ the strength of the nonlinear lateral-growth term, and $\eta$ Gaussian white noise (Kardar, Parisi, Zhang, Phys. Rev. Lett. 56, 889 (1986)). Setting $\lambda=0$ gives the linear Edwards–Wilkinson (EW) equation. Models sharing the asymptotic scaling of these equations form the KPZ and EW universality classes, characterized by the Family–Vicsek scaling of the global interface width (root-mean-square roughness) $W\sim t^{\beta}$ during growth and $W\sim L^{\alpha}$ at saturation on a substrate of linear size $L$, with dynamic exponent $z=\alpha/\beta$; in the KPZ class the Galilean (tilt) invariance enforces $\alpha+z=2$. In $1+1$ dimensions the KPZ exponents ($\beta=1/3$, $z=3/2$) and the full one-point height-fluctuation distributions (Tracy–Widom laws, i.e., the eigenvalue-fluctuation distributions of random-matrix ensembles) are known exactly. In $d\geq 2$ no exact results exist.

A central unresolved issue is the upper critical dimension $d_u$ of the KPZ class: the dimension at which power-law roughening would give way to logarithmic scaling and above which interfaces are asymptotically flat with Gaussian, EW-like fluctuations. For the EW class one knows exactly $d_u=2$ (from $\alpha=(2-d)/2$). For KPZ, analytic approaches disagree: self-consistent and mode-coupling theories give $d_u\lesssim 4$ (J.-P. Bouchaud and M. E. Cates, Phys. Rev. E 47, R1455 (1993); F. Colaiori and M. A. Moore, Phys. Rev. Lett. 86, 3946 (2001)), nonperturbative functional renormalization group (RG) studies indicate $d_u>4$ (T. Kloss, L. Canet, N. Wschebor, Phys. Rev. E 86, 051124 (2012); L. Canet, J. Stat. Mech. (2025) 124003), while other RG analyses argue $d_u=\infty$, i.e., the KPZ strong-coupling fixed point exists at every finite dimension (C. Castellano, M. Marsili, L. Pietronero, Phys. Rev. Lett. 80, 3527 (1998); C. Castellano et al., Phys. Rev. E 58, R5209 (1998)). Numerically, if $d_u$ is finite it exceeds 6 (S. G. Alves, T. J. Oliveira, S. C. Ferreira, Phys. Rev. E 90, 020103(R) (2014)), and no crossover has been detected in simulations reaching $d=15$ (T. J. Oliveira, Phys. Rev. E 106, L062103 (2022), which proposed the empirical exponent formula $\beta_d=7/(8d+13)$, positive at all finite $d$).

Because $d=\infty$ cannot be reached on hypercubic lattices, treelike and complete substrates are used as mean-field proxies. A Cayley tree (CT) is a finite tree in which every interior site has a fixed coordination number $q$; the Bethe lattice (BL) is the infinite core of such a tree, infinitely far from its boundary. The distinction is decisive: on a finite CT the boundary sites remain a finite fraction of all sites even as the tree grows, so CT observables are dominated by boundary effects, whereas equilibrium models such as the Ising model are solved exactly on the BL by working in the boundary-free core (the Bethe approximation). Distances on a tree are usually measured as the chemical distance (number of bonds along the unique path between two sites), but how such distances relate to the continuum dynamical exponents is not established. Motivated by this, Saberi (EPL 103, 10005 (2013)) simulated standard growth models on CTs — random deposition with surface relaxation (RDSR; a deposition rule with local downslope relaxation, in the EW class), restricted solid-on-solid (RSOS; deposition forbidding overhangs and large steps, in the KPZ class), and ballistic deposition (BD; particles stick to the first column they touch on arrival, a nonrelaxational KPZ rule) — reporting anomalous roughness and concluding that the KPZ nonlinearity stays relevant even at $d=\infty$. Oliveira (Europhys. Lett. 133, 28001 (2021)) showed these findings were artifacts of the CT boundary: interfaces grow curved, and global roughness is ill-defined on nonflat surfaces. Measuring instead the one-point height fluctuation $w_0$ at the central site — argued to approximate the BL — both classes show mild, essentially size-independent fluctuations, consistent with smooth surfaces expected for $d\geq d_u$; on complete graphs (all-to-all coupling, another standard infinite-dimensional limit) the RDSR and RSOS rules trivially grow layer-by-layer, while the behavior of BD could not be predicted. A related exactly solvable benchmark exists in the directed-polymer formulation of KPZ: on disordered trees the free energy of the directed polymer is exactly solvable with a traveling-wave distribution (B. Derrida, H. Spohn, J. Stat. Phys. 51, 817 (1988)) — but no analogous exact solution exists for growth-dynamics height statistics on the Bethe lattice, and it is not known how such a solution could be constructed.

Problem Statement

Establish the infinite-dimensional (mean-field) limit of KPZ-class surface growth and decide whether the KPZ universality class has a finite upper critical dimension, in the concrete setting left open by Oliveira (EPL 133, 28001 (2021)) after correcting the Cayley-tree boundary artifacts. The problem has one scientific objective — the fate of the KPZ nonlinearity as the substrate dimension grows — with the following jointly supporting deliverables, all stated by the source:

1. Determine whether the EW-like, asymptotically smooth behavior observed for KPZ-class models at the central site of Cayley trees (and, per 2026 preprints, for the continuum KPZ equation on complete graphs) sets in already at some finite spatial dimension $d_u$ — and if so, compute $d_u$ and the crossover scaling, including any logarithmic corrections — or whether the KPZ strong-coupling regime persists at every finite $d$ with EW-like behavior only in the $d\to\infty$ limit. An answer may be a rigorous theorem, a controlled renormalization-group construction, or an exact solution from which the high-dimensional scaling follows.

2. Develop the exact-solvability machinery for stochastic growth dynamics on the Bethe lattice — in the way equilibrium models such as the Ising model are solved exactly on the Bethe lattice via the boundary-free core of the infinite tree — for example, an exact solution for the height statistics (growth and/or steady state) of an EW- or KPZ-class growth model on the Bethe lattice, thereby placing the finite-Cayley-tree central-site approximation on exact footing.

3. Resolve the supporting ambiguities on which any such solution rests: (a) give a principled definition of distance and correlation measures on Cayley trees and their relation to the continuum dynamical exponents (e.g., whether the observed crossover time $t_c\sim k^2$ in trees of $k$ generations reflects a dynamic exponent $z=2$); (b) characterize the behavior of nonrelaxational KPZ rules such as ballistic deposition on effectively infinite-dimensional substrates (for complete graphs this was addressed in 2026 preprints, which found BD outside both universality classes; the Bethe-lattice behavior and the status of other nonrelaxational rules remain open); (c) establish whether central-site one-point diagnostics capture the universal height-distribution properties (full distributions and higher cumulants) of growth models in the Bethe-lattice limit, which 2025–2026 numerical work suggests they may not, given model-dependent non-Gaussian fluctuations.

The completion boundary of the problem is a determination of the fate of the KPZ nonlinearity in high and infinite dimension — a value of $d_u$ (finite, with its crossover behavior) or a demonstration that $d_u=\infty$ with the corresponding asymptotic scaling — ideally anchored by an exact Bethe-lattice (or equivalent mean-field) solution of at least one growth model. Deliverables 2 and 3 are the source's stated route to deliverable 1 and are jointly sufficient with it; an answer resolving deliverable 1 by any sound method, with deliverables 2 and 3 clarified to the extent needed to support it, completes the problem.

References

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