The $k$-XORSAT threshold revisited

TL;DR

This paper offers a simplified proof of the satisfiability threshold for random k-XORSAT problems and extends results to determine the full rank threshold for sparse random matrices over finite fields, combining physics-inspired methods with moment calculations.

Contribution

It provides a simplified proof of the k-XORSAT threshold and determines the full rank threshold for sparse matrices, extending prior work with new analytical techniques.

Findings

Simplified proof of the k-XORSAT satisfiability threshold.

Determination of the full rank threshold for sparse matrices.

Integration of message passing and moment methods.

Abstract

We provide a simplified proof of the random $k$-XORSAT satisfiability threshold theorem. As an extension we also determine the full rank threshold for sparse random matrices over finite fields with precisely $k$ non-zero entries per row. This complements a result from [Ayre, Coja-Oghlan, Gao, M\"uller: Combinatorica 2020]. The proof combines physics-inspired message passing arguments with a surgical moment computation.