  For a large class of repulsive interaction models, the Mayer cluster
integrals can be transformed into a tridiagonal real symmetric matrix $R_{mn}$,
whose elements converge to two constants. This allows for an effective
extrapolation of the equation of state for these models. Due to a nearby
(nonphysical) singularity on the negative real z axis, standard methods (e.g.
Pad\`e approximants based on the cluster integrals expansion) fail to capture
the behavior of these models near the ordering transition, and, in particular,
do not detect the critical point. A recent work (Eisenberg and Baram, PNAS {\bf
104}, 5755 (2007)) has shown that the critical exponents $\sigma$ and
$\sigma'$, characterizing the singularity of the density as a function of the
activity, can be exactly calculated if the decay of the $R$ matrix elements to
their asymptotic constant follows a $1/n^2$ law. Here we employ renormalization
arguments to extend this result and analyze cases for which the asymptotic
approach of the $R$ matrix elements towards their limiting value is of a more
general form. The relevant asymptotic correction terms (in RG sense) are
identified and we then provide a corrected exact formula for the critical
exponents. We identify the limits of usage of the formula, and demonstrate one
physical model which is beyond its range of validity. The new formula is
validated numerically and then applied to analyze a number of concrete physical
models.
