  We consider the (discrete) parabolic Anderson model $\partial u(t,x)/\partial
t=\Delta u(t,x) +\xi_t(x) u(t,x)$, $t\geq 0$, $x\in \mathbb{Z}^d$, where the
$\xi$-field is $\mathbb{R}$-valued and plays the role of a dynamic random
environment, and $\Delta$ is the discrete Laplacian. We focus on the case in
which $\xi$ is given by a properly rescaled symmetric simple exclusion process
under which it converges to an Ornstein--Uhlenbeck process. Scaling the
Laplacian diffusively and restricting ourselves to a torus, we show that in
dimension $d=3$ upon considering a suitably renormalised version of the above
equation, the sequence of solutions converges in law.
  As a by-product of our main result we obtain precise asymptotics for the
survival probability of a simple random walk that is killed at a scale
dependent rate when meeting an exclusion particle. Our proof relies on the
discrete theory of regularity structures of \cite{ErhardHairerRegularity} and
on novel sharp estimates of joint cumulants of arbitrary large order for the
exclusion process. We think that the latter is of independent interest and may
find applications elsewhere.
