Critical phenomena at edges and corners

Using Monte Carlo techniques, the critical behaviour at edges and corners of the three-dimensional Ising model is studied. In particular, the critical exponent $\beta_2$ of the local magnetization at edges formed by two intersecting free surfaces is estimated to be, as a function of the opening angle $\theta$, $0.96 \pm 0.02$ for $\theta = 135^o$, $1.28 \pm 0.04$ for $90^o$, and $2.30 \pm 0.10$ for $45^o$. The critical exponent $\beta_3$ of the corner magnetization of a cube is found to be $1.86 \pm 0.06$. The Monte Carlo estimates are compared to results of mean field theory, renormalization group calculations and high temperature series expansions.