  We examine a double trace deformation of SU(N) Yang-Mills theory which, for
large $N$ and large volume, is equivalent to unmodified Yang-Mills theory up to
$O(1/N^2)$ corrections. In contrast to the unmodified theory, large $N$ volume
independence is valid in the deformed theory down to arbitrarily small volumes.
The double trace deformation prevents the spontaneous breaking of center
symmetry which would otherwise disrupt large $N$ volume independence in small
volumes. For small values of $N$, if the theory is formulated on $\R^3 \times
S^1$ with a sufficiently small compactification size $L$, then an analytic
treatment of the non-perturbative dynamics of the deformed theory is possible.
In this regime, we show that the deformed Yang-Mills theory has a mass gap and
exhibits linear confinement. Increasing the circumference $L$ or number of
colors $N$ decreases the separation of scales on which the analytic treatment
relies. However, there are no order parameters which distinguish the small and
large radius regimes. Consequently, for small $N$ the deformed theory provides
a novel example of a locally four-dimensional pure gauge theory in which one
has analytic control over confinement, while for large $N$ it provides a simple
fully reduced model for Yang-Mills theory. The construction is easily
generalized to QCD and other QCD-like theories.
