in dimension $2$ to contain more than $O(n)$ elements. We will construct examples in the case when the time to travel over each vertex is from the set $\{1,2\}$ and when $M=+\infty$.

We will consider the subgraph $V_n= [-n,n]\times[0,n]$ of $\mathbb Z^2$. At time $0$ the water is located in all vertices of the $x$ axis. For sufficiently large $n$ we will provide an example of configuration $\omega$ of passage times for the edges of the graph $V_n$ such that the number of active vertices at time $n$ is of order $n\log n$. This would establish a lower bound on the probability that the number of active vertices at time $t$ is large.

Let us assume that each edge of the graph has the time component assigned from the set $\{1,2\}$ independently from each other. Assume that the probability that $1$ is assigned to each edge is equal to $p$, where $0<p<1$.

\[lower\_bound\] There exists $t_0\geq 0$, $\mu>0$, and $\alpha>0$ such that for each $t>t_0$ there exists $n$ such that the number $A_t$ of active vertices at time $t$ in the graph $V_n$ satisfies $$\mathbb P\left(A_t\geq \alpha t\log t\right)\geq e^{-\mu t^2}.$$

To prepare for the proof of the theorem we 