The Laplacian and $\bar\partial$ operators on critical planar graphs

TL;DR

This paper explores the properties of Laplacian and d-bar operators on critical planar graphs, providing explicit formulas for their determinants and inverses, and relating these to geometric and physical models.

Contribution

It introduces explicit formulas for determinants and inverses of operators on critical planar graphs and links these to hyperbolic geometry and statistical models.

Findings

Determinants depend only on local geometry.

Explicit formulas for inverses and determinants are derived.

Connections to hyperbolic volume and mean curvature are established.

Abstract

On a periodic planar graph whose edge weights satisfy a certain simple geometric condition, the discrete Laplacian and d-bar operators have the property that their determinants and inverses only depend on the local geometry of the graph. We obtain explicit expressions for the logarithms of the (normalized) determinants, as well as the inverses of these operators. We relate the logarithm of the determinants to the volume plus mean curvature of an associated hyperbolic ideal polyhedron. In the associated dimer and spanning tree models, for which the determinants of d-bar and the Laplacian respectively play the role of the partition function, this allows us to compute the entropy and correlations in terms of the local geometry. In addition, we define a continuous family of special discrete analytic functions, which, via convolutions gives a general process for constructing discrete analytic functions and discrete harmonic functions on critical planar graphs.