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We discuss a new formulation of the linear theory of discrete complex analysis on planar quad-graphs based on their medial graphs. It generalizes the theory on rhombic quad-graphs developed by Duffin, Mercat, Kenyon, Chelkak and Smirnov and follows the approach on general quad-graphs proposed by Mercat. We provide discrete counterparts of the most fundamental objects in complex analysis such as holomorphic functions, differential forms, derivatives, and the Laplacian. Also, we discuss discrete versions of important fundamental theorems such as Green’s identities and Cauchy’s integral formulae. For the first time, Green’s first identity and Cauchy’s integral formula for the derivative of a holomorphic function are discretized. 1. History Discrete harmonic functions on the square lattice were studied by a number of authors in the 1920s, including Courant, Friedrichs, and Lewy [5]. Discrete holomorphic functions on the square lattice were studied by Isaacs [10]. He proposed two different definitions for holomorphicity. One of them was reintroduced and further investigated by Lelong-Ferrand [8]. She developed the theory to a level that allowed her to prove the Riemann mapping theorem using discrete methods [13]. Duffin also studied discrete complex analysis on the square grid [6], and he was the first who extended the theory to rhombic lattices [7]. Kenyon [12], and Chelkak and Smirnov [3] resumed the investigation of discrete complex analysis on rhombic lattices, or, equivalently, isoradial graphs. Mercat extended the theory from domains in the complex plane to discrete Riemann surfaces, first considering cellular decompostions into rhombi [14] and later generalizing the notions to general quadrilaterals [16]. The motivation for this theory of discrete Riemann surfaces is derived from statistical physics, in particular, the Ising model. Mercat defined a discrete Dirac operator and discrete spin structures, and he identifies criticality in the Ising model with rhombic quadgraphs. Some two-dimensional discrete models in statistical physics exhibit conformally invariant properties in the thermodynamical limit. Such conformally invariant properties were established by Chelkak and Smirnov for the Ising model [4], and by Kenyon for the dimer model on a square grid [11]. In both cases, linear theories of discrete analytic functions on regular grids were highly important. Kenyon, Chelkak and Smirnov obtained important analytic results [12, 3], which were instrumental in the proof that the critical Ising model is universal [4]. Important non-linear discrete theories of complex analysis involve circle packings, or, more generally, circle patterns. Rodin and Sullivan first proved that the Riemann mapping of a complex domain to the unit disk can be approximated by circle packings [17]. A similar result for isoradial Text presented during the meeting “Discrete curvature: Theory and applications” organized by Laurent Najman and Pascal Romon. 18-22 novembre 2013, C.I.R.M. (Luminy). 2000 Mathematics Subject Classification. 39A12, 30G25.

Discrete complex analysis – the medial graph approach