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Pluri-Lagrangian systems are variational systems with the multi-dimensionalconsistency property. This notion has its roots in the theory of pluriharmonicfunctions, in the Z-invariant models of statistical mechanics, in the theory ofvariational symmetries going back to Noether and in the theory of discreteintegrable systems. A $d$-dimensional pluri-Lagrangian problem can be describedas follows: given a $d$-form $L$ on an $m$-dimensional space, $m > d$, whosecoefficients depend on a function $u$ of $m$ independent variables (calledfield), find those fields $u$ which deliver critical points to the actionfunctionals $S_\Sigma=\int_\Sigma L$ for any $d$-dimensional manifold $\Sigma$in the $m$-dimensional space. We investigate discrete 2-dimensional linearpluri-Lagrangian systems, i.e. those with quadratic Lagrangians $L$. The actionis a discrete analogue of the Dirichlet energy, and solutions are calleddiscrete pluriharmonic functions. We classify linear pluri-Lagrangian systemswith Lagrangians depending on diagonals. They are described by generalizationsof the star-triangle map. Examples of more general quadratic Lagrangians arealso considered.

Discrete pluriharmonic functions as solutions of linear pluri-Lagrangian systems