Critical and Topological Properties of Cluster Boundaries in the $3d$ Ising Model

TL;DR

This paper investigates the topological and critical properties of cluster boundaries in the 3D Ising model at criticality, revealing universal behaviors and a branching instability akin to non-critical string theories.

Contribution

It provides a detailed analysis of the genus and surface area distribution of critical cluster boundaries, demonstrating universality across different lattice types and connecting to string theory concepts.

Findings

Number of surfaces scales as a power law with area

Surface volume is proportional to surface area

Universal behavior observed across lattice types

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.