On the most compact regular lattice in large dimensions: A statistical mechanical approach

TL;DR

This paper applies a statistical mechanics approach to determine the maximum density of regular lattices in high dimensions, exploring connections with sphere packing entropy and presenting new conjectures.

Contribution

It introduces a novel statistical mechanics framework for lattice density optimization and highlights the relevance of Roger's theorems in this context.

Findings

Identifies similarities between lattice density and sphere packing entropy.

Proposes new conjectures relating lattice structures to liquid entropy.

Highlights the potential of statistical mechanics in high-dimensional lattice problems.

Abstract

In this paper I will approach the computation of the maximum density of regular lattices in large dimensions using a statistical mechanics approach. The starting point will be some theorems of Roger, which are virtually unknown in the community of physicists. Using his approach one can see that there are many similarities (and differences) with the problem of computing the entropy of a liquid of perfect spheres. The relation between the two problems is investigated in details. Some conjectures are presented, that need further investigation in order to check their consistency.