  This paper studies the problem of clustering in the two-component Gaussian
mixture model where the centers are separated by $2\Delta$ for some $\Delta>0$.
We characterize the exact phase transition threshold, given by $$
\bar{\Delta}_n^{2} = \sigma^{2}\left(1 + \sqrt{1+\frac{2p}{n\log{n}}}
\right)\log{n}, $$ such that perfect recovery of the communities is possible
with high probability if $\Delta\ge(1+\varepsilon)\bar \Delta_n$, and
impossible if $\Delta\le (1-\varepsilon)\bar \Delta_n$ for any constant
$\varepsilon>0$. This implies an elbow effect at a critical dimension
$p^{*}=n\log{n}$.
  We present a non-asymptotic lower bound for the corresponding minimax Hamming
risk improving on existing results. It is, to our knowledge, the first lower
bound capturing the right dependence on $p$. We also propose an optimal,
efficient and adaptive procedure that is minimax rate optimal. The rate
optimality is moreover sharp in the asymptotics when the sample size goes to
infinity. Our procedure is based on a variant of Lloyd's iterations initialized
by a spectral method; a popular clustering algorithm widely used by
practitioners. Numerical studies confirm our theoretical findings.
