On lattices with finite Coulombian interaction energy in the plane

TL;DR

This paper investigates the properties of lattices in the plane with finite Coulombian interaction energy, providing insights into their structure and energy characteristics.

Contribution

It introduces a detailed analysis of the renormalized energy for planar lattices with Coulomb interactions, extending previous definitions and exploring conditions for finite energy.

Findings

Identification of conditions for finite Coulombian energy in planar lattices

Extension of the renormalized energy definition to broader lattice classes

Insights into the interaction energy behavior in two-dimensional lattice systems

Abstract

Given a discrete set $\Lambda$ in the plane (we will also say a {\em lattice}) and a real number $m\ge 0$, the renormalized energy introduced in \cite{ss1} heuristically describes the interaction energy of unit charges placed at the points of $\Lambda$ with a uniform negative background of density $m\in\R$. It is defined in several steps, following mostly \cite{ss1}.