  We show that the chiral partition function of two-dimensional Yang-Mills
theory on the sphere can be mapped to the partition function of the homogeneous
six-vertex model with domain wall boundary conditions in the ferroelectric
phase. A discrete matrix model description in both cases is given by the
Meixner ensemble, leading to a representation in terms of a stochastic growth
model. We show that the partition function is a particular case of the
z-measure on the set of Young diagrams, yielding a unitary matrix model for
chiral Yang-Mills theory on the sphere and the identification of the partition
function as a tau-function of the Painleve V equation. We describe the role
played by generalized non-chiral Yang-Mills theory on the sphere in relating
the Meixner matrix model to the Toda chain hierarchy encompassing the
integrability of the six-vertex model. We also argue that the thermodynamic
behaviour of the six-vertex model in the disordered and antiferroelectric
phases are captured by particular q-deformations of two-dimensional Yang-Mills
theory on the sphere.
