Bures--Wasserstein Covariance Geometry in a Gaussian Concentration Regime

We study the covariance geometry associated with probability measures in a
Gaussian concentration regime. Near a nondegenerate minimum of a smooth
potential, Laplace asymptotics identify a local Gaussian approximation, so
that the quadratic Wasserstein geometry of the concentrating family is
asymptotically described by the geometry of Gaussian measures. On the
Gaussian manifold, the quadratic Wasserstein distance is exact and, in the
centred covariance sector, reduces to the Bures--Wasserstein metric on the
cone of positive-definite matrices. The corresponding intrinsic
gradient-flow structure on the centred Gaussian submanifold is therefore
finite-dimensional and takes place on covariance space.

The main result of the paper concerns the internal geometry of this
covariance sector in even dimension. Using Williamson's symplectic
decomposition, we consider the local orbit generated by independent one-mode
symplectic transformations and compute the Bures--Wasserstein orthogonal
complement of its tangent space at a Williamson normal form. We show that
this orthogonal complement decomposes explicitly into radial directions,
which change the symplectic eigenvalues, and off-diagonal directions, which
create cross-mode covariance couplings. Consequently, if a smooth functional
is critical along the local orbit, then its Bures--Wasserstein gradient lies
in this transverse space; if the radial coefficients vanish, the gradient is
purely off-diagonal.

We also record the behaviour of the Bures--Wasserstein metric near the
boundary of the positive-semidefinite cone: in the classical concentration
regime, the covariance approaches the rank-deficient boundary, where the
metric becomes singular in transverse directions. A brief final remark
explains how the same covariance variables admit a quantum Gaussian
interpretation, but the main metric results of the paper are entirely
classical.