Basic techniques in two-dimensional critical Ising percolation with investigation of scaling relations

TL;DR

This paper investigates the scaling relations in two-dimensional critical Ising percolation, demonstrating that most relations hold under certain power law assumptions, with one notable exception.

Contribution

It establishes the validity of all but one hyperscaling relation in 2D Ising percolation under specific assumptions.

Findings

Most scaling relations hold under power law assumptions.

One hyperscaling relation does not hold.

Results apply to high-temperature Ising models near critical fields.

Abstract

We consider the percolation problem in the high-temperature Ising model on the two-dimensional square lattice at/near critical external fields. We show that all scaling relations, except a single hyperscaling relation, hold under the power law assumptions for the one-arm path and four-arm paths.