It looks like Joseph O'Rourke has beaten me, but I've generated a bit more data than he has:

Let $l$ be the lower bound so that the non-diagonal entries are uniformly distributed between $l$ and 1 and the diagonal entries are distributed between $l+1$ and 2.  Note that Piero's original characterization of his code was inaccurate, but I just used his code to generate the figures below.

Here is a grid of eigenvalues of such $N$ by $N$ matrices with $l=-.9$ (plots are labeled by $N$)

![-.9 plot][2]

Here's the picture with $l=-.87$

![-.87 plot][1]

Here's the picture with $l=-.93$

![-.93 plot][3]

And here's a picture of the absolute value of the largest eigenvalue of the matrix as a function of $N$ (now the label is $l$)

![plot of largest eigenvalues][4]

**Edit based on Helge's comment**: Here's a picture of $\max_{1\leq j \leq N} \sum_{n=1}^{N} u_j^N(n)$ as a function of $N$ (the $u_j^N$ are normalized eigenvectors) with $l=-0.9$

![plot of largest sum][5]


And here's a picture of the further normalized version $\max_{1\leq j \leq N} \frac{1}{\sqrt{N}}\sum_{n=1}^{N} u_j^N(n)$

![plot of largest sum normalized][6]

  [1]: https://i583.photobucket.com/albums/ss275/jaspercrowne/Screenshot2010-07-03at93508AM.png
  [2]: https://i583.photobucket.com/albums/ss275/jaspercrowne/Screenshot2010-07-03at92431AM.png
  [3]: https://i583.photobucket.com/albums/ss275/jaspercrowne/Screenshot2010-07-03at92844AM.png
  [4]: https://i583.photobucket.com/albums/ss275/jaspercrowne/Screenshot2010-07-03at95101AM.png
  [5]: https://i583.photobucket.com/albums/ss275/jaspercrowne/Screenshot2010-07-03at103302AM.png
  [6]: https://i583.photobucket.com/albums/ss275/jaspercrowne/Screenshot2010-07-03at103813AM.png