Consider zombies placed uniformly at random over $\mathbb{R}^2$ with asymptotic density $\mu$ zombies/area. You are placed at a random point and can move with speed $1$. Zombies move with speed $v\leq 1$ straight towards you, what is the probability $P(\mu,v)$ you can escape to infinity without a zombie catching you?

What if the zombies can move in any direction and might collude to set up a wall of high density or similar tactics?

Lets call it a win for zombies if for every $d>0$, one of them can get within a distance $d$ in finite time.

Addendum: Is there a finite collection of colluding zombies and a player placement, from which escape is impossible? What is the least number of zombies?