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

Abstract. Two discretizations, linear and nonlinear, of basic notions of the complex analysis are considered. The underlying lattice is an arbitrary quasicrystallic rhombic tiling of a plane. The linear theory is based on the discrete Cauchy-Riemann equations, the nonlinear one is based on the notion of circle patterns. We clarify the role of the rhombic condition in both theories: under this condition the corresponding equations are integrable (in the sense of 3D consistency, which yields also the existense of zero curvature representations, Bäcklund transformations etc.). We demonstrate that in some precise sense the linear theory is a linearization of the nonlinear one: the tangent space to a set of integrable circle patterns at an isoradial point consists of discrete holomorphic functions which take real (imaginary) values on two sublattices. We extend solutions of the basic equations of both theories to Z d , where d is the number of different edge slopes of the quasicrystallic tiling. In the linear theory, we give an integral representation of an arbitrary discrete holomorphic function, thus proving the density of discrete exponential functions. We introduce the d-dimensional discrete logarithmic function which is a generalization of Kenyon's discrete Green's function, and uncover several new properties of this function. We prove that it is an isomonodromic solution of the discrete Cauchy-Riemann equations, and that it is a tangent vector to the space of integrable circle patterns along the family of isomonodromic discrete power functions.

We construct integrable lattice realizations of conformal twisted boundary conditions for sℓ(2) unitary minimal models on a torus. These conformal field theories are realized as the continuum scaling limit of critical A-D-E lattice models with positive spectral parameter. The integrable seam boundary conditions are labelled by (r, s, ζ) ∈ (A g−2 , A g−1 , Γ) where Γ is the group of automorphisms of the graph G and g is the Coxeter number of G = A, D, E. Taking symmetries into account, these are identified with conformal twisted boundary conditions of Petkova and Zuber labelled by (a, b, γ) ∈ (A g−2 ⊗G, A g−2 ⊗G, Z 2 ) and associated with nodes of the minimal analog of the Ocneanu quantum graph. Our results are illustrated using the Ising (A 2 , A 3 ) and 3-state Potts (A 4 , D 4 ) models.

We detail the theory of Discrete Riemann Surfaces. It takes place on a cellular decomposition of a surface, together with its Poincaré dual, equipped with a discrete conformal structure. A lot of theorems of the continuous theory follow through to the discrete case, we will define the discrete analogs of period matrices, Riemann's bilinear relations, exponential of constant argument and series. We present the notion of criticality and its relationship with integrability.