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

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.