  Theaimofthepresentpaperistosuggestthatstatisticalphysicsprovides the correct
language to understand the practical behavior of the LLL algorithm, most of
which are left unexplained to this day. To this end, we propose sandpile models
that imitate LLL with compelling accuracy, and prove for these models some of
the most desired statements regarding LLL. We also formulate a few conjectures
that formally capture our heuristics and would serve as milestones for further
development of the theory.
