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Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green’s function. (English) Zbl 1099.37054

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.

MSC:

37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
30G25 Discrete analytic functions
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