Superdiffusive Central Limit Theorem for the Stochastic Burgers Equation at the critical dimension

TL;DR

This work resolves the long-standing question of the large-scale behavior of the 2d Stochastic Burgers Equation at the critical dimension by proving a sharp, logarithmically superdiffusive diffusion with explicit constants and a Gaussian fixed point under a logarithmically adjusted scaling. The authors develop a resolvent-based Renormalization Group framework, introducing a novel diagonal/off-diagonal decomposition and Replacement Lemmas to control the generator on Wiener chaos across all scales. They establish a precise diffusion coefficient $D(t) \sim C_{eff}(\lambda) (\log t)^{2/3}$ with $C_{eff}(\lambda) = (\tfrac{3}{2\pi})^{2/3} \lambda^{4/3}$ and show convergence to an effective anisotropic linear stochastic heat equation with diffusion matrix $D_{eff}$. The results provide a first rigorous scaling limit for a critical singular SPDE beyond weak coupling and suggest universality for other critical driven diffusive systems, with broad implications for stochastic turbulence and related particle systems.

Abstract

The Stochastic Burgers Equation (SBE) is a singular, non-linear Stochastic Partial Differential Equation (SPDE) that describes, on mesoscopic scales, the fluctuations of stochastic driven diffusive systems with a conserved scalar quantity. In space dimension d = 2, the SBE is critical, being formally scale invariant under diffusive scaling. As such, it falls outside of the domain of applicability of the theories of Regularity Structures and paracontrolled calculus. In apparent contrast with the formal scale invariance, we fully prove the conjecture first appeared in [H. van Beijeren, R. Kutner, & H. Spohn, Phys. Rev. Lett., 1986] according to which the 2d-SBE is logarithmically superdiffusive, i.e. its diffusion coefficient diverges like $(\log t)^{2/3}$ as $t\to\infty$, thus removing subleading diverging multiplicative corrections in [D. De Gaspari & L. Haunschmid-Sibitz, Electron. J. Probab., 2024] and in [H.-T. Yau, Ann. of Math., 2004] for 2d-ASEP. We precisely identify the constant prefactor of the logarithm and show it is proportional to $λ^{4/3}$, for $λ>0$ the coupling constant, which, intriguingly, turns out to be exactly the same as for the one-dimensional Stochastic Burgers/KPZ equation. More importantly, we prove that, under super-diffusive space-time rescaling, the SBE has an explicit Gaussian fixed point in the Renormalization Group sense, by deriving a superdiffusive central limit-type theorem for its solution. This is the first scaling limit result for a critical singular SPDE, beyond the weak coupling regime, and is obtained via a refined control, on all length-scales, of the resolvent of the generator of the SBE. We believe our methods are well-suited to study other out-of-equilibrium driven diffusive systems at the critical dimension, such as 2d-ASEP, which, we conjecture, have the same large-scale Fixed Point as SBE.

Theorems & Definitions (60)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 1.5
• Definition 1.6
• Lemma 1.7
• Definition 2.1
• Remark 2.2
• Proposition 2.3