  Linear lattice gauge theory is based on link variables that are arbitrary
complex or real $N\times N$ matrices, in distinction to the usual (non-linear)
formulation with unitary or orthogonal matrices. For a large region in
parameter space both formulations belong to the same universality class, such
that the continuum limits of linear and non-linear lattice gauge theory are
identical. We explore if the linear formulation can help to find a
non-perturbative continuum limit formulated in terms of continuum fields.
Linear lattice gauge theory exhibits excitations beyond the gauge fields. In
the linear formulation the running gauge coupling corresponds to the flow of
the minimum of a ``link potential''. This minimum occurs for a nonzero value of
the link variable $l_0$ in the perturbative regime, while $l_0$ vanishes in the
confinement regime. We discuss a flow equation for the scale dependent location
of the minimum $l_0(k)$.
