The fugitive is at the origin. They move at a speed of $1$. There's a guard at $(i,j)$ for all $i,j\in \mathbb{Z}$ except the origin. A guard's speed is $\frac{1}{100}$. The fugitive and the guards move simultaneously and continuously. At any moment, all guards move towards the current position of the fugitive, i.e. a guard's trajectory is a [pursuit curve](https://en.wikipedia.org/wiki/Pursuit_curve). If they're within $\frac{1}{100}$ distance from a guard, the fugitive is caught. The game is played on $\mathbb{R}^2$.

[![enter image description here][1]][1]

Question: can the fugitive avoid capture forever?

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What I know:

 1. The fugitive will be caught if they remain in a bounded area.

 
 2. The distance between two guards is always non-increasing, and is strictly decreasing unless the fugitive and the guards remain collinear. But the further away the guards are, the slower that distance decreases.   

 3. If there're only 2 guards, the fugitive will be caught if they make a straight-line dash into the gap between the guards, as long as the guards are sufficiently far away (see [radiodrome](https://en.wikipedia.org/wiki/Radiodrome)). But the fugitive can always find a non-straight path to safely slip through between them.

 4. The fugitive can escape if they're enclosed by a wall of guards that's not too "thick" (4 or 5 layers are fine), such as this (3 layers)

[![enter image description here][2]][2]


The shape (concentric circles or polygons) and size of the wall don't change the outcome. 


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Source: I asked the question sometime ago on [mathexchange](https://math.stackexchange.com/questions/4127588/escaping-infinitely-many-pursuers), where I received some really cool [animation](https://i.sstatic.net/uG0xp.gif) by martin. I was inspired by a very [similar problem](https://mathoverflow.net/questions/177364/escape-the-zombie-apocalypse) here on MO, with additional complication of randomness.    


  [1]: https://i.sstatic.net/zFVqNm.jpg
  [2]: https://i.sstatic.net/qKIcgm.png