  Let $(G,\rho)$ be a stationary random graph, and use $B^G_{\rho}(r)$ to
denote the ball of radius $r$ about $\rho$ in $G$. Suppose that $(G,\rho)$ has
annealed polynomial growth, in the sense that $\mathbb{E}[|B^G_{\rho}(r)|] \leq
O(r^k)$ for some $k > 0$ and every $r \geq 1$.
  Then there is an infinite sequence of times $\{t_n\}$ at which the random
walk $\{X_t\}$ on $(G,\rho)$ is at most diffusive: Almost surely (over the
choice of $(G,\rho)$), there is a number $C > 0$ such that \[ \mathbb{E}
\left[\mathrm{dist}_G(X_0, X_{t_n})^2 \mid X_0 = \rho, (G,\rho)\right]\leq C
t_n\qquad \forall n \geq 1\,. \] This result is new even in the case when $G$
is a stationary random subgraph of $\mathbb{Z}^d$. Combined with the work of
Benjamini, Duminil-Copin, Kozma, and Yadin (2015), it implies that $G$ almost
surely does not admit a non-constant harmonic function of sublinear growth.
  To complement this, we argue that passing to a subsequence of times $\{t_n\}$
is necessary, as there are stationary random graphs of (almost sure) polynomial
growth where the random walk is almost surely superdiffusive at an infinite
subset of times.
