  Recent advances in the study of conformally invariant discrete random
processes have lead to increasing interest in the study of discrete analogues
to holomorphic functions. Of particular interest are results which provide
conditions under which these discrete functions can be shown to converge to
continuum versions as the lattice spacing shrinks to zero. Recent work by
Skopenkov has extended these results to include a wide class of non-uniform
quadrilateral lattices with a pair of regularity conditions, one local and one
global.
  Such a result is sufficient for the study of random processes on
deterministic lattices, however to establish convergence results for
conformally invariant random processes on random triangulations, such a global
regularity condition cannot be assumed. In this paper we provide a convergence
result on quadrilateral lattices upon which we enforce only a local condition
on the geometry of each face.
