Diffusive scaling limit in dimension four

Establish, for the four-dimensional divisible sandpile with initial masses \(\sigma=1+8\zeta\), where \((\zeta(x))_{x\in\mathbb{Z}^4}\) are i.i.d. mean-zero Gaussian variables with positive variance, the limiting law in \(H^{-s}_{\mathrm{loc}}(\mathbb{R}^4)\) of the diffusively rescaled centered odometer fields \(\bigl(u_{\lfloor TR^2\rfloor}-E u_{\lfloor TR^2\rfloor}(0)\bigr)^{(R)}\) for every \(T>0\) and \(s>0\).

Background

In dimension four, the paper proves tightness of the centered odometer at diffusive times in every negative local Sobolev space, but identifies a membrane-model limit only at superdiffusive times t=⌊Rα⌋t=\lfloor R^\alpha\rfloor with α>2\alpha>2. The diffusive regime t≍R2t\asymp R^2 is therefore not resolved.

The open problem asks both for convergence at diffusive times and for identification of the limiting random distribution. Its difficulty comes from the nonlinear optimal-stopping structure of the odometer and the logarithmic correlations of the associated four-dimensional membrane field.

References

Such a scale is unavailable when $t\asymp R2$; in that regime we prove tightness in every negative Sobolev space but do not identify the limit. Instead we prove a weaker, superdiffusive scaling limit: at times $t=\lfloor R\alpha\rfloor$ with $\alpha>2$ the intermediate scale is available, and after subtracting a smooth spatial average, $u_t-E u_t(0)$ converges to the four-dimensional continuum membrane model $\mathcal G_4$, modulo constants. The diffusive limit remains open (Problem~\ref{prob:d4-diffusive}).

— Quantitative explosion and percolation of the divisible sandpile  (2609.02829 - Bou-Rabee et al., 2 Sep 2026) in Problem 1.2, Section 1.3.2; reiterated in Section 5, subsection “Scaling limit in dimension four”