  We measure the running of the $SU(\infty)$ 't Hooft coupling by performing a
step scaling analysis of the Twisted Eguchi-Kawai (TEK) model, the SU($N$)
gauge theory on a single site lattice with twisted boundary conditions. The
computation relies on the conjecture that finite volume effects for SU(N) gauge
theories defined on a 4-dimensional twisted torus are controlled by an
effective size parameter $\tilde l = l \sqrt{N}$, with $l$ the torus period. We
set the scale for the running coupling in terms of $\tilde l$ and use the
gradient flow to define a renormalized 't Hooft coupling $\lambda(\tilde l)$.
In the TEK model, this idea allows the determination of the running of the
coupling through a step scaling procedure that uses the rank of the group as a
size parameter. The continuum renormalized coupling constant is extracted in
the zero lattice spacing limit, which in the TEK model corresponds to the large
$N$ limit taken at fixed value of $\lambda(\tilde l)$. The coupling constant is
thus expected to coincide with that of the ordinary pure gauge theory at $N
=\infty$. The idea is shown to work and permits us to follow the evolution of
the coupling over a wide range of scales. At weak coupling we find a remarkable
agreement with the perturbative two-loop formula for the running coupling.
