  Recently we introduced a family of U(N) invariant random matrix ensembles
which is a one-paramter ($\lambda$) extension of the q-random matrix ensembles
(RMEs), given by the asymptotic weak confining potential $V(H) \sim [\ln
H]^{(1+\lambda)}$ \cite{cm-jpa09}. With numerical construction of the
corresponding orthogonal polynomials, we showed that the eigenvalue density of
the ensembles deviates from the inverse power law and that the two-level kernel
of the ensembles is qualitatively different from those of Gaussian and the
critical ensembles. In this work, we make further efforts to characterize the
two-level kernel of the $\lambda$-ensembles and discuss its various properties.
To this end, we first show that the kernel of the $\lambda$-ensembles also
possess an anomalous structure characteristic of the critical ensembles, namely
the ghost correlation peak. We then propose, albeit in a restricted regime, a
form of the two-level kernel which is distinct from the sine kernel of the
Gaussian ensembles as well as the sinh kernel of the critical ensembles. We
test the proposed form numerically and discuss its implications. In particular,
we show that the case $\lambda > 1$ is qualitatively distinct from the case
$\lambda <1$, the latter decribing true fat-tail ensembles.
