  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.
