Violating conformal invariance: Two-dimensional clusters grafted to wedges, cones, and branch points of Riemann surfaces

TL;DR

This paper investigates the behavior of 2D lattice animals grafted onto complex geometries, revealing conformal invariance only at small angles and proposing a heuristic explanation for large angles, with implications for critical phenomena.

Contribution

The study provides the first detailed simulation analysis of 2D lattice animals on non-trivial geometries, demonstrating angle-dependent scaling behaviors and extending understanding of conformal invariance violations.

Findings

Conformal invariance holds for small angles ($ heta o 1/ ext{angle}$).

At large angles, the entropic exponent approaches a constant minus a linear term in angle.

Comparison with critical percolation highlights differences in geometric scaling behaviors.

Abstract

We present simulations of 2-d site animals on square and triangular lattices in non-trivial geomeLattice animals are one of the few critical models in statistical mechanics violating conformal invariance. We present here simulations of 2-d site animals on square and triangular lattices in non-trivial geometries. The simulations are done with the newly developed PERM algorithm which gives very precise estimates of the partition sum, yielding precise values for the entropic exponent $\theta$ ($Z_N \sim \mu^N N^{-\theta}$). 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, $\theta \sim 1/\alpha$, only for small angles ($\alpha \ll 2\pi$), while $\theta \approx const -\alpha/2\pi$ for $\alpha \gg 2\pi$. These scalings hold both for wedges and cones. A heuristic (non-conformal) argument for the behavior at large $\alpha$ is given, and comparison is made with critical percolation.