  Recently Tracy and Widom conjectured [math.CO/9904042] and Johansson proved
[math.CO/9906120] that the expected shape \lambda of the semi-standard tableau
produced by a random word in k letters is asymptotically the spectrum of a
random traceless k by k GUE matrix. In this article we give two arguments for
this fact. In the first argument, we realize the random matrix itself as a
quantum random variable on the space of random words, if this space is viewed
as a quantum state space. In the second argument, we show that the distribution
of \lambda is asymptotically given by the usual local limit theorem, but the
resulting Gaussian is disguised by an extra polynomial weight and by reflecting
walls. Both arguments more generally apply to an arbitrary finite-dimensional
representation V of an arbitrary simple Lie algebra g. In the original
question, V is the defining representation of g = su(k).
