Geometrical organization of solutions to random linear Boolean equations

TL;DR

This paper investigates the geometric structure of solutions in large random linear Boolean systems, revealing how solutions cluster and identifying thresholds where solutions become disconnected.

Contribution

It provides a detailed analysis of the solution space geometry for random XORSAT problems using the cavity method, including the distribution of cluster distances and satisfiability thresholds.

Findings

Distribution of distances between solution clusters computed

Identification of the x-satisfiability threshold

Insights into the clustering phase transition

Abstract

The random XORSAT problem deals with large random linear systems of Boolean variables. The difficulty of such problems is controlled by the ratio of number of equations to number of variables. It is known that in some range of values of this parameter, the space of solutions breaks into many disconnected clusters. Here we study precisely the corresponding geometrical organization. In particular, the distribution of distances between these clusters is computed by the cavity method. This allows to study the `x-satisfiability' threshold, the critical density of equations where there exist two solutions at a given distance.