  We consider the problem of Gaussian mixture clustering in the
high-dimensional limit where the data consists of $m$ points in $n$ dimensions,
$n,m \rightarrow \infty$ and $\alpha = m/n$ stays finite. Using exact but
non-rigorous methods from statistical physics, we determine the critical value
of $\alpha$ and the distance between the clusters at which it becomes
information-theoretically possible to reconstruct the membership into clusters
better than chance. We also determine the accuracy achievable by the
Bayes-optimal estimation algorithm. In particular, we find that when the number
of clusters is sufficiently large, $r > 4 + 2 \sqrt{\alpha}$, there is a gap
between the threshold for information-theoretically optimal performance and the
threshold at which known algorithms succeed.
