NPAC Technical Report SCCS-621
Critical and Topological Properties of Cluster Boundaries in the 3d Ising Model
V. Dotsenko, G. Harris, E. Marinari, E. Marinec, M. Picco, P. Windey
Submitted April 1 1994
Abstract
We analyze the behavior of the ensemble of surface boundaries of the
critical clusters at $T=T_c$ in the $3d$ Ising model. We find that
$N_g(A)$, the number of surfaces of given genus $g$ and fixed area
$A$, behaves as $A^{-x(g)}$ $e^{-\mu A}$. We show that $\mu$ is a
constant independent of $g$ and $x(g)$ is approximately a linear
function of $g$.
The sum of $N_g(A)$ over genus scales as a power of $A$.
We also observe that the volume of the clusters is
proportional to its surface area. We argue that this behavior is
typical of a branching instability for the surfaces, similar to the
ones found for non-critical string theories with $c > 1$. We discuss
similar results for the ordinary spin clusters of the $3d$ Ising model
at the minority percolation point and for $3d$ bond percolation.
Finally we check the universality of these critical properties on the
simple cubic lattice and the body centered cubic lattice.