In addition to the paper arXiv:1301.2222 by Burago et al we can look in the following questions:

• work by S. Smirnov on discrete Cauchy-Riemann equations: Discrete Complex Analysis and Probability, Discrete complex analysis and probability (ICM 2010) and Discrete Complex Analysis
• work of Giné and Koltchinskii: Empirical graph Laplacian approximation of Laplace–Beltrami operators: Large sample results
• using random walks and stochastic analysis to approximate the Laplacians on groups, such as the hypoelliptic Laplacian on the Heisenberg group, following Gordina and Laetsch Sub-Laplacians on Sub-Riemannian Manifolds