time*]{}, i.e. the first time at which the vertex $x$ is reached.

The active vertices at time $t$ are located near the boundary of the ball $B_t$. It is known that for large $t$ the set $\frac1tB_t$ will be approximately convex. More precisely, it is known [@cox_durrett] that there is a convex set $B$ such that $$\mathbb P\left((1-\varepsilon)B\subseteq \frac1tB_t\subseteq (1+\varepsilon)B \mbox{ for large }t \right)=1.$$

However, the previous theorem does not guarantee that the boundary of $B_t$ has to be of zero volume. In fact the boundary can be non-polygonal as was previously shown [@damron_hochman].

The set of active vertices does not coincide with the boundary of $B_t$, but it is expected that if $\partial B_t$ is of small volume then the number of active vertices is small in most typical configurations of random graphs. We provide an example for which the set of active vertices is a large fractal, but simulations suggest that this does not happen in average scenario.

The fluctuations of the shape of $B_t$ are expected to be of order $t^{2/3}$ in the case of $\mathbb Z^2$ and the first passage time $\tau(0,n)$ is proven to have fluctuations of order at least $\log n$ [@newman_piza]. The 