  We introduce a new random matrix model called distance covariance matrix in
this paper, whose normalized trace is equivalent to the distance covariance. We
first derive a deterministic limit for the eigenvalue distribution of the
distance covariance matrix when the dimensions of the vectors and the sample
size tend to infinity simultaneously. This limit is valid when the vectors are
independent or weakly dependent through a finite-rank perturbation. It is also
universal and independent of the details of the distributions of the vectors.
Furthermore, the top eigenvalues of this distance covariance matrix are shown
to obey an exact phase transition when the dependence of the vectors is of
finite rank. This finding enables the construction of a new detector for such
weak dependence where classical methods based on large sample covariance
matrices or sample canonical correlations may fail in the considered
high-dimensional framework.
