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Discrete Pluriharmonic Functions as Solutions of Linear Pluri-Lagrangian Systems

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Abstract

Pluri-Lagrangian systems are variational systems with the multi-dimensional consistency property. This notion has its roots in the theory of pluriharmonic functions, in the Z-invariant models of statistical mechanics, in the theory of variational symmetries going back to Noether and in the theory of discrete integrable systems. A d-dimensional pluri-Lagrangian problem can be described as follows: given a d-form L on an m-dimensional space, m > d, whose coefficients depend on a function u of m independent variables (called field), find those fields u which deliver critical points to the action functionals \({S_\Sigma=\int_\Sigma L}\) for any d-dimensional manifold Σ in the m-dimensional space. We investigate discrete 2-dimensional linear pluri-Lagrangian systems, i.e., those with quadratic Lagrangians L. The action is a discrete analogue of the Dirichlet energy, and solutions are called discrete pluriharmonic functions. We classify linear pluri-Lagrangian systems with Lagrangians depending on diagonals. They are described by generalizations of the star-triangle map. Examples of more general quadratic Lagrangians are also considered.

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Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany

    A. I. Bobenko & Yu. B. Suris

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  1. A. I. Bobenko
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  2. Yu. B. Suris
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Correspondence to Yu. B. Suris.

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Communicated by N. Reshetikhin

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Bobenko, A.I., Suris, Y.B. Discrete Pluriharmonic Functions as Solutions of Linear Pluri-Lagrangian Systems. Commun. Math. Phys. 336, 199–215 (2015). https://doi.org/10.1007/s00220-014-2240-5

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