Percolation in the three-dimensional Ising model

TL;DR

This study investigates percolation phenomena in the three-dimensional Ising model, revealing only one transition in 3D versus two in 2D, and explores layered systems showing unique critical exponents.

Contribution

It demonstrates the absence of double percolation transitions in 3D Ising models and characterizes critical exponents for layered systems, extending understanding of geometric critical phenomena.

Findings

Only one percolation transition in 3D Ising configurations.

Layered systems exhibit distinct critical exponents and universality class.

Theoretical analysis supports simulation results for the complete graph.

Abstract

Geometric representations provide a useful perspective on critical phenomena in the Ising model. In a recent study [Phys. Rev. E 112, 034118 (2025)], we found that the two-dimensional critical Ising model exhibits two consecutive percolation transitions for geometric spin clusters as the bond-occupation probability $p$ between parallel spins increases. Here, through extensive Monte Carlo simulations, we show that this phenomenon does not persist in three dimensions, where we observe only a single percolation transition on critical Ising configurations. Further theoretical analysis of the Ising model on the complete graph also yields the same scenario. In addition, we study percolation on a two-dimensional layer embedded in the three-dimensional critical Ising model. For this layer system, we estimate the red-bond exponent $y_p = 0.426(6)$ and the fractal dimensions of the largest cluster, hull, and shortest path as $d_f = 1.8926(20)$, $d_{\rm hull} = 1.663(4)$, and $d_{\rm min} = 1.080(10)$, respectively. These values indicate a distinct universality class induced by coupling to out-of-plane critical correlations.