High-precision anomalous dimension of $3d$ percolation from giant cluster slicing

TL;DR

This paper uses a geometric approach to precisely determine the anomalous dimension of 3D percolation, providing more accurate results than previous measurements and testing hyperscaling relations.

Contribution

The study introduces a novel application of the critical geometry approach to 3D percolation, deriving the order parameter profile from the fractional Yamabe equation to obtain the anomalous dimension.

Findings

Anomalous dimension η = -0.0431(14) with high precision

Order parameter profile shape related to fractional Yamabe equation

Hyperscaling relation tested and confirmed

Abstract

We apply the critical geometry approach for bounded critical phenomena [1] to $3d$ percolation. The functional shape of the order parameter profile $\phi$ is related via the fractional Yamabe equation to its scaling dimension $\Delta_{\phi}$. We obtain $\Delta_{\phi}= 0.4785(7)$ from which the anomalous dimension $\eta$ is found to be $\eta=-0.0431(14)$, a value compatible with, and more precise than, its previous direct measurements. A test of hyperscaling is also performed.