Inside the clustering window for random linear equations

TL;DR

This paper investigates the structure of solutions in random linear equations over GF(2), revealing a smooth transition in cluster connectivity around the known clustering threshold, extending understanding of phase transitions in such problems.

Contribution

It extends the analysis of clustering in r-XORSAT to the near-threshold regime, showing a smooth transition in cluster connectivity parameters.

Findings

Connectivity parameters undergo a smooth transition around the clustering threshold

Solutions form a single connected cluster below the threshold and multiple well-separated clusters above

The transition is continuous, not abrupt, near the threshold

Abstract

We study a random system of cn linear equations over n variables in GF(2), where each equation contains exactly r variables; this is equivalent to r-XORSAT. Previous work has established a clustering threshold, c^*_r for this model: if c=c_r^*-\epsilon for any constant \epsilon>0 then with high probability all solutions form a well-connected cluster; whereas if c=c^*_r+\epsilon, then with high probability the solutions partition into well-connected, well-separated clusters (with probability tending to 1 as n goes to infinity). This is part of a general clustering phenomenon which is hypothesized to arise in most of the commonly studied models of random constraint satisfaction problems, via sophisticated but mostly non-rigorous techniques from statistical physics. We extend that study to the range c=c^*_r+o(1), and prove that the connectivity parameters of the r-XORSAT clusters undergo a smooth transition around the clustering threshold.