Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1993-04-21
Phys.Rev.Lett. 71 (1993) 811-814
Physics
High Energy Physics
High Energy Physics - Theory
REVISED NOW NO APS MACROS NEEDED FOR HEPTH USERS 4 p 4 f, EFI 93-24
Scientific paper
10.1103/PhysRevLett.71.811
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.
Dotsenko Victor
Harris Geoffrey
Marinari Enzo
Martinec Emil
Picco Marco
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