It seems the most reasonable way to formalize the problem is saying that at start the zombies are distributed according to a Poisson process in the plane with density $\mu$. As this is (distributionally) translation invariant we can assume that you start at the origin.

Now we observe that changing the zombie configuration in any finite box will not affect the outcome. This tells us the event "getting caught" belongs to the tail $\sigma$-fleld. So by Kolmogorov's 0-1 law the probability is either 0 or 1.

So it suffices to show said probability is positive. As pointed out by other readers it is easy to see that there are ways to position zombies at a short distance from the human as to guarantee capture in a short time. As such configurations have positive probability, we are done.

If zombies are allowed to have a strategy then Joseph Van Name's answer already tells the whole story.

It seems the most reasonable way to formalize the problem is saying that at start the zombies are distributed according to a Poisson process in the plane with density $\mu$. As this is (distributionally) translation invariant we can assume that you start at the origin.

Now we observe that changing the zombie configuration in any finite box will not affect the outcome. This tells us the event "getting caught" belongs to the tail $\sigma$-fleld. So by Kolmogorov's 0-1 law the probability is either 0 or 1.

[EDIT] I wrote "So it suffices to show said probability is positive. As pointed out by other readers it is easy to see that there are ways to position zombies at a short distance from the human as to guarantee capture in a short time. As such configurations have positive probability, we are done." This is wrong. You can indeed escape as pointed out by Pablo Lessa. You will be infinitely many times closer to a zombie than any given distance, but you will make it safely to infinity. [EDIT]

If zombies are allowed to have a strategy then Joseph Van Name's answer already tells the whole story.