  We study the asymptotic behavior of the partition function and the
correlation kernel in random matrix ensembles of the form $\frac{1}{Z_n}
\big|\det \big( M^2-tI \big)\big|^{\alpha} e^{-n\operatorname{Tr} V(M)}dM$,
where $M$ is an $n\times n$ Hermitian matrix, $\alpha>-1/2$ and $t\in\mathbb
R$, in double scaling limits where $n\to\infty$ and simultaneously $t\to 0$. If
$t$ is proportional to $1/n^2$, a transition takes place which can be described
in terms of a family of solutions to the Painlev\'e V equation. These
Painlev\'e solutions are in general transcendental functions, but for certain
values of $\alpha$, they are algebraic, which leads to explicit asymptotics of
the partition function and the correlation kernel.
