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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.