  The goal of this article is to rederive the connection between the Painlev\'e
$5$ integrable system and the universal eigenvalues correlation functions of
double-scaled hermitian matrix models, through the topological recursion
method. More specifically we prove, \textbf{to all orders}, that the WKB
asymptotic expansions of the $\tau$-function as well as of determinantal
formulas arising from the Painlev\'e $5$ Lax pair are identical to the large
$N$ double scaling asymptotic expansions of the partition function and
correlation functions of any hermitian matrix model around a regular point in
the bulk. In other words, we rederive the "sine-law" universal bulk asymptotic
of large random matrices and provide an alternative perturbative proof of
universality in the bulk with only algebraic methods. Eventually we exhibit the
first orders of the series expansion up to $O(N^{-5})$.
