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

Discrete analytic functions on non-uniform lattices without global geometric control