  We describe a novel technique that renders theories of $N$ axions tractable,
and more generally can be used to efficiently analyze a large class of periodic
potentials of arbitrary dimension. Such potentials are complex energy
landscapes with a number of local minima that scales as $\sqrt{N!}$, and so for
large $N$ appear to be analytically and numerically intractable. Our method is
based on uncovering a set of approximate symmetries that exist in addition to
the $N$ periods. These approximate symmetries, which are exponentially close to
exact, allow us to locate the minima very efficiently and accurately and to
analyze other characteristics of the potential. We apply our framework to
evaluate the diameters of flat regions suitable for slow-roll inflation, which
unifies, corrects and extends several forms of "axion alignment" previously
observed in the literature. We find that in a broad class of random theories,
the potential is smooth over diameters enhanced by $N^{3/2}$ compared to the
typical scale of the potential. A Mathematica implementation of our framework
is available online.

High Energy Physics - Theory