Bobenko, Alexander I.; Mercat, Christian; Suris, Yuri B. Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green’s function. (English) Zbl 1099.37054 J. Reine Angew. Math. 583, 117-161 (2005). The authors deal with the linear and nonlinear theories of discrete analytic functions theory, showing in some precise sense the former is a linearization of the latter. Here, they work in the set-up of rhombic tilings of a plane. They show that a discrete Cauchy-Riemann equations on a rhombically embedded quad-graph \({\mathcal M}\), with weights given by quotients of diagonals of the corresponding rhombic, are integrable. They also show that, cross-ratio equations on a rhombically embedded quad-graph \({\mathcal M}\), with cross-ratios read off the corresponding rhombic, are integrable as well. Therefore, solutions of the cross-ratio equations on a quasicrystal rhombing embedding \({\mathcal M}\) are naturally extended to \(\mathbb Z^d\).The authors define discrete exponential functions on \(\mathbb Z^d\), and prove that they are dense in the space of discrete holomorphic functions, growing not faster than exponentially. They introduce the \(d\)-dimensional discrete logarithm function which is a generalization of Kenyou’s discrete Green’s function, and uncover several new properties of this function. Moreover, the authors prove that it is an isomonodromic solution of the discrete Cauchy-Riemann equations, and that it is a tangent vector to the space of integrable circle patterns along the family of isomonodromic discrete power functions. Reviewer: Messoud A. Efendiev (Berlin) Cited in 38 Documents MSC: 37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry 30G25 Discrete analytic functions Keywords:discrete analytic functions; integrable structure; isomonodromic Green’s function; Cauchy-Riemann equations PDFBibTeX XMLCite \textit{A. I. Bobenko} et al., J. Reine Angew. Math. 583, 117--161 (2005; Zbl 1099.37054) Full Text: DOI arXiv