Inside the clustering window for random linear equations

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, cr∗ for this model: if c=cr∗−ϵ for any constant ϵ>0 then with high probability all solutions form a well‐connected cluster; whereas if c=cr∗+ϵ, then with high probability the solutions partition into well‐connected, well‐separated clusters (with probability tending to 1 as n→∞). 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 nonrigorous techniques from statistical physics. We extend that study to the range c=cr∗+o(1), and prove that the connectivity parameters of the r‐XORSAT clusters undergo a smooth transition around the clustering threshold.

The stripping process can be slow: Part I

“…The aforementioned bounds on the depth and stripping number from were motivated by applications to solution clustering in random XORSAT. We will provide analogous applications for our present results in a subsequent paper .…”

“…The aforementioned bounds on the depth and stripping number from [1,21] were motivated by applications to solution clustering in random XORSAT. We will provide analogous applications for our present results in a subsequent paper (a preliminary version containing partial results on their applications to clustering in random XORSAT is available in [17]).…”

“…The set R ( v ) would still have contained a stripping sequence leading to the removal of v . We define, and bound, this larger set for our application to clusters in random XOR‐SAT in another paper .Observation R ( v ) contains a stripping sequence ending with v .Proof We prove this for each using a simple induction on i . For i = 1, v is a stripping sequence of length 1.…”

“…Remark For the purposes of this paper, we could have omitted step (a) and replaced R � by R elsewhere. The set R(v) would still have contained a stripping sequence leading to the removal of v. We define, and bound, this larger set for our application to clusters in random XOR-SAT in another paper [15].…”

“…The set R(v) would still have contained a stripping sequence leading to the removal of v. We define, and bound, this larger set for our application to clusters in random XOR-SAT in another paper [16]. (A preliminary version is in [17]).…”