Discrete Pluriharmonic Functions as Solutions of Linear Pluri-Lagrangian Systems

A. I. Bobenko, Yu B. Suris

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14 Scopus citations


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 (RFormula presented.) 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.

Original languageEnglish
Pages (from-to)199-215
Number of pages17
JournalCommunications in Mathematical Physics
Issue number1
StatePublished - May 2015
Externally publishedYes


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