  We propose a new family of matrix models whose 1/N expansion captures the
all-genus topological string on toric Calabi-Yau threefolds. These matrix
models are constructed from the trace class operators appearing in the
quantization of the corresponding mirror curves. The fact that they provide a
non-perturbative realization of the (standard) topological string follows from
a recent conjecture connecting the spectral properties of these operators, to
the enumerative invariants of the underlying Calabi-Yau threefolds. We study in
detail the resulting matrix models for some simple geometries, like local P^2
and local F_2, and we verify that their weak 't Hooft coupling expansion
reproduces the topological string free energies near the conifold singularity.
These matrix models are formally similar to those appearing in the Fermi-gas
formulation of Chern-Simons-matter theories, and their 1/N expansion receives
non-perturbative corrections determined by the Nekrasov-Shatashvili limit of
the refined topological string.
