  We consider the detection problem of correlations in a $p$-dimensional
Gaussian vector, when we observe $n$ independent, identically distributed
random vectors, for $n$ and $p$ large. We assume that the covariance matrix
varies in some ellipsoid with parameter $\alpha >1/2$ and total energy bounded
by $L>0$. We propose a test procedure based on a U-statistic of order 2 which
is weighted in an optimal way. The weights are the solution of an optimization
problem, they are constant on each diagonal and non-null only for the $T$ first
diagonals, where $T=o(p)$. We show that this test statistic is asymptotically
Gaussian distributed under the null hypothesis and also under the alternative
hypothesis for matrices close to the detection boundary. We prove upper bounds
for the total error probability of our test procedure, for $\alpha>1/2$ and
under the assumption $T=o(p)$ which implies that $n=o(p^{ 2 \alpha})$. We
illustrate via a numerical study the behavior of our test procedure. Moreover,
we prove lower bounds for the maximal type II error and the total error
probabilities. Thus we obtain the asymptotic and the sharp asymptotically
minimax separation rate $\tilde{\varphi} = (C(\alpha, L) n^2 p )^{- \alpha/(4
\alpha + 1)}$, for $\alpha>3/2$ and for $\alpha >1$ together with the
additional assumption $p= o(n^{4 \alpha -1})$, respectively. We deduce rate
asymptotic minimax results for testing the inverse of the covariance matrix. We
construct an adaptive test procedure with respect to the parameter $\alpha$ and
show that it attains the rate $\tilde{\psi}= ( n^2 p / \ln\ln(n
\displaystyle\sqrt{p}) )^{- \alpha/(4 \alpha + 1)}$.
