NPAC Technical Report SCCS-486
Critical and Topological Properties of Cluster Boundaries in the 3-D Ising Model
V Dotsenko, P Windey, G Harris, E Marinari, E Martinec, M Picco
Submitted April 21 1993
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.