Some empirical data, with each point the average of 50 trials:
<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<img src="http://cs.smith.edu/~orourke/MathOverflow/MinDistSquare.jpg" alt="MinDistSquare" />
<br />
The curve is the best-fit $a + b/n$.  $(a,b)=(-0.01,0.99)$.