Inside the clustering threshold for random linear equations

TL;DR

This paper investigates the structure of solutions in random linear equations over GF(2), revealing how solution clusters become connected or separated as the system approaches the clustering threshold, with implications for understanding the geometry of solution spaces.

Contribution

It extends the understanding of the clustering phenomenon in random linear systems near the threshold, quantifying the connectivity within and between solution clusters.

Findings

Solution connectivity within clusters scales as n^{Θ(δ)}

Moving between clusters requires at least n^{1-O(δ)} variable changes

Vertex removal in hypergraphs near the k-core threshold also scales as n^{Θ(δ)}

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. \cite{ikkm,amxor} determined the clustering threshold, $c^*_r$: if $c=c^*_r+\e$ for any constant $\e>0$, then \aas the solutions partition into well-connected, well-separated {\em clusters} (with probability tending to 1 as $n\rightarrow\infty$). 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 $c=c^*_r+o(1)$, showing that if $c=c^*_r+n^{-\d}, \d>0$, then the connectivity parameter of each $r$-XORSAT cluster is $n^{\Theta(\d)}$, as compared to $O(\log n)$ when $c=c^*_r+\e$. This means that one can move between any two solutions in the same cluster via a sequence of solutions where consecutive solutions differ on at most $n^{\Theta(\d)}$ variables; this is tight up to the implicit constant. In contrast, moving to a solution in another cluster requires that some pair of consecutive solutions differ in at least $n^{1-O(\d)}$ variables. Along the way, we prove that in a random $r$-uniform hypergraph with edge-density $n^{-\d}$ above the $k$-core threshold, \aas every vertex not in the $k$-core can be removed by a sequence of $n^{\Theta(\d)}$ vertex-deletions in which the deleted vertex has degree less than $k$; again, this is tight up to the implicit constant.