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

The sine-law gap probability, Painlev\'e 5, and asymptotic expansion by the topological recursion