  We study the large N expansion of a family of matrix models related to
topological strings on toric Calabi-Yau threefolds. These matrix models compute
spectral observables of underlying operators obtained by quantizing the mirror
curves. They have the form of a deformed O(2) matrix model, with a specific
non-polynomial potential involving the Faddeev quantum dilogarithm. Their
planar limit is studied using a particular conformal mapping depending on two
parameters, from which several universal results can be obtained. As expected,
the spectral curves controlling the planar limit of the matrix models are the
mirror curves themselves, which in our cases have genus 1. Our results
encompass all those toric geometries with genus $1$ mirror where an explicit
one-cut matrix integral is known: local $P^2$, local $F_0$, local $F_2$, and
degenerations of the resolved $C^3/Z_5$, the resolved $C^3/Z_6$ and the
resolved $Y^{3,0}$ geometries amongst others.
