Dimension-dependent symmetry breaking in a diffusive nonlocal interaction model

We study a diffusive nonlocal interaction equation generated by the squared volumes of random $n$-simplices. The nonlinear drift depends on the solution only through its mean and covariance matrix, which yields a closed finite-dimensional covariance system and an explicit representation of the measure-valued solution as an affine pushforward followed by Gaussian convolution. We use this reduction to prove global well-posedness for arbitrary initial data in ${\mathcal{P}}_2({\mathbb{R}}^d)$. We then classify the stationary states and show that they coincide with the minimizers of the associated free energy. When $1\le n<d$, the equilibrium is a unique isotropic Gaussian up to translation. By contrast, at the critical dimension $n=d$, the energy fixes only the determinant of the covariance matrix, and, when $d\ge 2$, a continuum of anisotropic Gaussian equilibria appears. Finally, we prove convergence of every solution to one of these equilibria in quadratic Wasserstein distance and obtain explicit exponential convergence rates.