Universal diagonal estimates for minimizers of the Levy-Lieb functional

Given a wave-function minimizing the Levy-Lieb functional, the intent of this short note is to give an estimate of the probability of the particles being in positions $(x_1,\dots ,x_N)$ in the $\delta$-close regime $D_{\delta }={\cup }_{i\mathrm{\ne }j}\{\mathrm{\mid }{x}_i-x_j\mathrm{\mid }\le \delta \}$.