  We discuss the perturbative expansion of SU(N) Yang-Mills theories defined on
a d-dimensional torus of linear size l with twisted boundary conditions,
generalizing previous results in the literature. For a specific class of twist
tensors depending on a single integer flux value k, we show that perturbative
results to all orders depend on the combination lN^(2/d) and a flux-dependent
angle theta. This implies a new kind of volume independence that holds at
finite N and for fixed values of theta. Our results also provide interesting
information about the possible occurrence of tachyonic instabilities at
one-loop order. We support the prescription that instabilities are avoided, if
the large N limit is taken keeping theta > theta_c, and appropriately scaling
the magnetic flux k with N. Numerical results in 2+1 dimensions provide a test
of how these ideas extend into the non-perturbative regime.
