Discrete Riemann Surfaces and the Ising Model

Abstract

A bstract: W e de ne a new theory of di screte R i em ann surfaces and present i ts basi c resul ts.T he key i dea i s to consi der not onl y a cel l ul ar decom posi ti on ofa surface,but the uni on w i th i ts dual .D i screte hol om orphy i s de ned by a strai ghtforward di screti sati on oftheC auchy-R i em ann equati on.A l otofcl assi cal resul tsi n R i em ann theory have a di screte counterpart,H odge star,harm oni ci ty, H odge theorem ,W eyl ' s l em m a,C auchy i ntegralform ul a,exi stence ofhol om orphi c form s w i th prescri bed hol onom i es. G i vi ng a geom etri cal m eani ng to the constructi on on a R i em ann surface,we de ne a noti on of cri ti cal i ty on w hi ch we prove a conti nuous l i m i t theorem .W e i nvesti gate i ts connecti on w i th cri ti cal i ty i n the Isi ng m odel .W e set up a D i rac equati on on a di screte uni versal spi n structure and we prove that the exi stence ofa D i rac spi nori sequi val entto cri ti cal i ty. C ontents

Smooth 2D Coordinate Systems on Discrete Surfaces

“…Following [13], we call this data a discrete conformal structure and we say that a parameterization is discrete conformal if it preserves the ratios ρ. In other words, for all faces of the mesh, we require that…”

“…In this paper we present a new algorithm to compute conformal parameterizations using the definition of discrete conformity given in [13]. Although, it was first used for meshes, with floating point coordinates, it has the main advantage of being easily adaptable to digital surfaces, with integer coordinates.…”

Rhombic embeddings of planar quad-graphs

“…Duffin's ideas were rediscovered by Mercat [6], [5] who used them to build up an extended theory of discrete holomorphy in one complex dimension.…”

“…Such embedding arises in discrete complex analysis [1], [5] and in statistical mechanics [3], [6], [4]. Here we study the spaces of such embeddings.…”

Functions on discrete sets holomorphic in the sense of isaacs, or monodiffric functions of the first kind

“…The set {z; Re z 0, Im z 0, Re z + Im z 3} {(1, 1)} is a domain of holomorphy in the sense of Definition 5.4, but not if we allow non-schlicht domains. A theory of holomorphic functions on the vertices and their dual vertices on a Riemann surface has been developed by Mercat [11,12] .…”