# Universal diagonal estimates for minimizers of the Levy-Lieb functional

## TL;DR

This paper provides estimates on the probability of particles being close together in the context of the Levy-Lieb functional, offering insights into the behavior of minimizers in quantum many-body systems.

## Contribution

It introduces universal diagonal estimates for minimizers of the Levy-Lieb functional, advancing understanding of particle proximity probabilities.

## Key findings

- Derived bounds on particle proximity probabilities
- Quantified likelihood of particles being within delta distance
- Enhanced understanding of minimizer behavior in quantum systems

## Abstract

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 $(x_1, \ldots, x_N)$ in the $\delta$-close regime $D_{\delta}= \cup_{i \neq j} \{|x_i - x_j| \leq \delta\}$.