The small-$N$ series in the zero-dimensional $O(N)$ model: constructive expansions and transseries

We consider the 0-dimensional quartic $O(N)$ vector model and present a complete study of the partition function $Z(g,N)$ and its logarithm, the free energy $W(g,N)$, seen as functions of the coupling $g$ on a Riemann surface. Using constructive field theory techniques we prove that both $Z(g,N)$ and $W(g,N)$ are Borel summable functions along all the rays in the cut complex plane $\mathbb{C}_{\pi} =\mathbb{C}\setminus \mathbb{R}_-$. We recover the transseries expansion of $Z(g,N)$ using the intermediate field representation. We furthermore study the small-$N$ expansions of $Z(g,N)$ and $ W(g,N)$. For any $g=|g| e^{\imath \varphi}$ on the sector of the Riemann surface with $|\varphi|