  The distribution of overlaps of solutions of a random CSP is an indicator of
the overall geometry of its solution space. For random $k$-SAT, nonrigorous
methods from Statistical Physics support the validity of the ``one step replica
symmetry breaking'' approach. Some of these predictions were rigorously
confirmed in \cite{cond-mat/0504070/prl} \cite{cond-mat/0506053}. There it is
proved that the overlap distribution of random $k$-SAT, $k\geq 9$, has
discontinuous support. Furthermore, Achlioptas and Ricci-Tersenghi proved that,
for random $k$-SAT, $k\geq 8$. and constraint densities close enough to the
phase transition there exists an exponential number of clusters of satisfying
assignments; moreover, the distance between satisfying assignments in different
clusters is linear.
  We aim to understand the structural properties of random CSP that lead to
solution clustering. To this end, we prove two results on the cluster structure
of solutions for binary CSP under the random model from Molloy (STOC 2002)
  1. For all constraint sets $S$ (described explicitly in Creignou and Daude
(2004), Istrate (2005)) s.t. $SAT(S)$ has a sharp threshold and all $q\in
(0,1]$, $q$-overlap-$SAT(S)$ has a sharp threshold (i.e. the first step of the
approach in Mora et al. works in all nontrivial cases). 2. For any constraint
density value $c<1$, the set of solutions of a random instance of 2-SAT form,
w.h.p., a single cluster. Also, for and any $q\in (0,1]$ such an instance has
w.h.p. two satisfying assignment of overlap $\sim q$. Thus, as expected from
Statistical Physics predictions, the second step of the approach in Mora et al.
fails for 2-SAT.
