  The Stochastic Burgers equation was introduced in [H. van Beijeren, R. Kutner
and H. Spohn, Excess noise for driven diffusive systems, PRL, 1985] as a
continuous approximation of the fluctuations of the asymmetric simple exclusion
process. It is formally given by $$\partial_t\eta =\frac{1}{2}\Delta\eta+
\mathfrak w\cdot\nabla(\eta^2) + \nabla\cdot\xi,$$ where $\xi$ is
$d$-dimensional space time white noise and $\mathfrak w$ is a fixed non-zero
vector. In the critical dimension $d=2$ at stationarity, we show that this
system exhibits superdiffusve behaviour: more specifically, its bulk diffusion
coefficient behaves like $(\log t)^\frac23$, in a Tauberian sense, up to
$\log\log\log t$ corrections. This confirms a prediction made in the physics
literature and complements [G. Cannizzarro, M. Gubinelli, F. Toninelli,
Gaussian Fluctuations for the stochastic Burgers equation in dimension $d\geq
2$, CMP, 2024], where the same equation was studied in the weak-coupling
regime. Furthermore this model can be seen as a continuous analogue to [H.T.
Yau, $(\log t)^\frac{2}{3}$ law of the two dimensional asymmetric simple
exclusion process, Annals of Mathematics, 2004].
