Logarithmically corrected Edwards–Wilkinson scaling for the directed exclusion process

Let $({\mathcal Y}_t^\varepsilon)_{t\ge 0}$ be the fluctuation field of the directed exclusion process, with $s_\varepsilon(t)$ determined by

$$t=\varepsilon^{2}s_\varepsilon(t)\log(s_\varepsilon(t))^{1/2}.$$

The field

$${\mathcal Y}_t^\varepsilon(f)=\varepsilon^{1/2}\sum_{x\in\mathbb Z}f(\varepsilon x-\varepsilon s_\varepsilon(t))\left(\eta_{s_\varepsilon(t)}(x)-\frac12\right)$$

for test functions $f:\mathbb R\to\mathbb R$ and $t\ge 0$ should converge, in the appropriate distributional space, to the infinite-dimensional Ornstein–Uhlenbeck process described by the Edwards–Wilkinson equation. The marginally relevant nonlinear term is expected to leave the limiting field unchanged while introducing the logarithmic correction to the time scaling; for fixed $t>0$, this gives $s_\varepsilon(t)\sim t\varepsilon^{-2}|\log\varepsilon|^{-1/2}$ as $\varepsilon\to0$.