A physical study of the LLL algorithm

TL;DR

This study explores the LLL algorithm through the lens of statistical physics, proposing a sandpile model analogy that explains its behavior and predicts its asymptotic properties.

Contribution

It introduces a novel connection between LLL and sandpile models, providing theoretical proofs and empirical validation for their universality and scaling behavior.

Findings

LLL and sandpile models share the same universality class.

Finite-size scaling formulas accurately describe LLL behavior.

Predicted asymptotic RHF average is approximately 1.02265.

Abstract

This paper presents a study of the LLL algorithm from the perspective of statistical physics. Based on our experimental and theoretical results, we suggest that interpreting LLL as a sandpile model may help understand much of its mysterious behavior. In the language of physics, our work presents evidence that LLL and certain 1-d sandpile models with simpler toppling rules belong to the same universality class. This paper consists of three parts. First, we introduce sandpile models whose statistics imitate those of LLL with compelling accuracy, which leads to the idea that there must exist a meaningful connection between the two. Indeed, on those sandpile models, we are able to prove the analogues of some of the most desired statements for LLL, such as the existence of the gap between the theoretical and the experimental RHF bounds. Furthermore, we test the formulas from the finite-size scaling theory (FSS) against the LLL algorithm itself, and find that they are in excellent agreement. This in particular explains and refines the geometric series assumption (GSA), and allows one to extrapolate various quantities of interest to the dimension limit. In particular, we predict the empirical average RHF converges to $\approx 1.02265$ as dimension goes to infinity.