  The Random Satisfiability problem has been intensively studied for decades.
For a number of reasons the focus of this study has mostly been on the model,
in which instances are sampled uniformly at random from a set of formulas
satisfying some clear conditions, such as fixed density or the probability of a
clause to occur. However, some non-uniform distributions are also of
considerable interest. In this paper we consider Random 2-SAT problems, in
which instances are sampled from a wide range of non-uniform distributions.
  The model of random SAT we choose is the so-called configuration model, given
by a distribution $\xi$ for the degree (or the number of occurrences) of each
variable. Then to generate a formula the degree of each variable is sampled
from $\xi$, generating several \emph{clones} of the variable. Then 2-clauses
are created by choosing a random paritioning into 2-element sets on the set of
clones and assigning the polarity of literals at random.
  Here we consider the random 2-SAT problem in the configuration model for
power-law-like distributions $\xi$. More precisely, we assume that $\xi$ is
such that its right tail $F_{\xi}(x)$ satisfies the conditions
$W\ell^{-\alpha}\le F_{\xi}(\ell)\le V\ell^{-\alpha}$ for some constants $V,W$.
The main goal is to study the satisfiability threshold phenomenon depending on
the parameters $\alpha,V,W$. We show that a satisfiability threshold exists and
is determined by a simple relation between the first and second moments of
$\xi$.
