LLL and stochastic sandpile models

TL;DR

This paper uses sandpile models inspired by statistical physics to better understand the behavior of the LLL algorithm, providing new insights and formal conjectures for future research.

Contribution

It introduces sandpile models that accurately imitate LLL and proves key properties, advancing the theoretical understanding of LLL's practical behavior.

Findings

Sandpile models effectively replicate LLL behavior

Proved several desired properties of LLL within these models

Formulated conjectures to guide future theoretical development

Abstract

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.