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.