  Using the technique of evolving sets, we explore the connection between
entropy growth and transience for simple random walks on connected infinite
graphs with bounded degree. In particular we show that for a simple random walk
starting at a vertex $x_0$, if the entropy after $n$ steps, $E_n$ is at least
$Cn$ where the $C$ is independent of $x_0$, then the random walk is transient.
We also give an example which demonstrates that the condition of $C$ being
independent of $x_0$ is necessary.
