title: Violating conformal invariance: two-dimensional clusters grafted to wedges, cones, and branch points of Riemann surfaces.
abstract: Lattice animals are one of the few critical models in statistical mechanics violating conformal invariance. We present here simulations of two-dimensional site animals on square and triangular lattices in nontrivial geometries. The simulations are done with the pruned-enriched Rosenbluth method (PERM) algorithm, which gives very precise estimates of the partition sum, yielding precise values for the entropic exponent theta (Z(N) approximately micro(N)N(-theta)). In particular, we studied animals grafted to the tips of wedges with a wide range of angles alpha, to the tips of cones (wedges with the sides glued together), and to branching points of Riemann surfaces. The latter can either have k sheets and no boundary, generalizing in this way cones to angles alpha>360 degrees, or can have boundaries, generalizing wedges. We find conformal invariance behavior, theta approximately 1/alpha , only for small angles (alpha << 2pi) , while theta approximately = const-alpha/2pi for alpha << 2pi. These scalings hold both for wedges and cones. A heuristic (nonconformal) argument for the behavior at large alpha is given, and comparison is made with critical percolation.
authors.affiliation: John-von-Neumann Institute for Computing, Forschungszentrum Jülich, D-52425 Jülich, Germany.
authors.forename: Hsiao-Ping
authors.initials: HP
authors.lastname: Hsu
authors.forename: Walter
authors.initials: W
authors.lastname: Nadler
authors.forename: Peter
authors.initials: P
authors.lastname: Grassberger