  Let $(Y,(X_i)_{i\in\mathcal{I}})$ be a zero mean Gaussian vector and $V$ be a
subset of $\mathcal{I}$. Suppose we are given $n$ i.i.d. replications of the
vector $(Y,X)$. We propose a new test for testing that $Y$ is independent of
$(X_i)_{i\in \mathcal{I}\backslash V}$ conditionally to $(X_i)_{i\in V}$
against the general alternative that it is not. This procedure does not depend
on any prior information on the covariance of $X$ or the variance of $Y$ and
applies in a high-dimensional setting. It straightforwardly extends to test the
neighbourhood of a Gaussian graphical model. The procedure is based on a model
of Gaussian regression with random Gaussian covariates. We give non asymptotic
properties of the test and we prove that it is rate optimal (up to a possible
$\log(n)$ factor) over various classes of alternatives under some additional
assumptions. Besides, it allows us to derive non asymptotic minimax rates of
testing in this setting. Finally, we carry out a simulation study in order to
evaluate the performance of our procedure.
