Pluripotential Energy

TL;DR

This paper introduces two functionals based on discrete measures and Vandermonde determinants, demonstrating their equivalence to the electrostatic energy of a measure in complex space, generalizing classical logarithmic energy.

Contribution

It establishes the equivalence of two new functionals with electrostatic energy, extending classical energy concepts from the complex plane to higher dimensions.

Findings

The functionals $J(mu)$ and $W(mu)$ coincide up to a constant.

This equivalence generalizes the classical logarithmic energy to multivariate settings.

The results connect discrete approximations with continuous energy in complex analysis.

Abstract

For probability measures $\mu$ on compact subsets of $\CC^n$ we define two functionals $J(\mu)$ and $W(\mu)$ modeled on discrete approximations to $\mu$ and multivariate Vandermonde determinants. We show that these functionals coincide, up to a constant, with the electrostatic energy of $\mu$ defined in a more general setting by Berman, Boucksom, Guedj and Zeriahi. This generalizes the classical notion of logarithmic energy of a measure in the complex plane; i.e., the case $n=1$.