# Discrete Riemann surfaces: linear discretization and its convergence

## M. B. Skopenkov

Slides are available by following the link.

We develop linear discretization of complex analysis, originally introduced by R. Isaacs, J. Ferrand, R. Duffin, and C. Mercat [2]. We prove convergence of discrete period matrices and discrete Abelian integrals to their continuous counterparts. We also prove a discrete counterpart of the Riemann-Roch theorem. The proofs use energy estimates inspired by electrical networks [3].

This is a joint work with A. Bobenko [1].

### References

[1] A. Bobenko, M. Skopenkov, Discrete Riemann surfaces: linear discretization and its convergence

Submitted - 2012 - http://arxiv.org/abs/1210.0561.

[2] C. Mercat, Discrete Riemann surfaces and the Ising model * Comm. Math. Phys. - 2001. - 218:1. - 177-216.*

[3] M. Skopenkov, Boundary value problem for discrete analytic functions

Submitted - 2011 - http://arxiv.org/abs/1110.6737.