Say I sample $n$ points uniformly at random in the unit square, and then I look for the shortest path through $\sqrt{n}$ of those points (rounding up, say). What happens to the length of this path as $n\rightarrow\infty$? Does it increase, decrease, or converge?

Say I sample $n$ points uniformly at random in the unit square, and then I look for the shortest path through $\sqrt{n}$ of those points (rounding up, say). What happens to the length of this path as $n\rightarrow\infty$? Does it increase, decrease, or converge (or "none of the above")?