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Geometric properties of satisfying assignments of random $ε$-1-in-k SAT

We study the geometric structure of the set of solutions of random $ε$-1-in-k SAT problem. For $l\geq 1$, two satisfying assignments $A$ and $B$ are $l$-connected if there exists a sequence of satisfying assignments connecting them by changing at most $l$ bits at a time. We first prove that w.h.p. two assignments of a random $ε$-1-in-$k$ SAT instance are $O(\log n)$-connected, conditional on being satisfying assignments. Also, there exists $ε_{0}\in (0,\frac{1}{k-2})$ such that w.h.p. no two satisfying assignments at distance at least $ε_{0}\cdot n$ form a "hole" in the set of assignments. We believe that this is true for all $ε>0$, and thus satisfying assignments of a random 1-in-$k$ SAT instance form a single cluster.