Mass scaling of the near-critical 2D Ising model using random currents

TL;DR

This paper provides a new proof for the mass scaling of the near-critical 2D Ising model at critical temperature with an external magnetic field, using the random current representation and recent couplings, confirming the mass scales as h^{8/15}.

Contribution

It introduces a novel proof of the exponential decay of correlations and mass scaling in the 2D Ising model using random currents, differing from previous CLE-based methods.

Findings

Mass is proportional to h^{8/15} as h approaches zero.

Established exponential decay of two-point correlations.

Connected the random current approach with recent couplings to the random cluster model.

Abstract

We examine the Ising model at its critical temperature with an external magnetic field $h a^{\frac{15}{8}}$ on $a\mathbb{Z}^2$ for $a,h >0$. A new proof of exponential decay of the truncated two-point correlation functions is presented. It is proven that the mass (inverse correlation length) is of the order of $h^\frac{8}{15}$ in the limit $h \to 0$. This was previously proven with CLE-methods in $\lbrack 1 \rbrack$. Our new proof uses instead the random current representation of the Ising model and its backbone exploration. The method further relies on recent couplings to the random cluster model $\lbrack 2 \rbrack$ as well as a near-critical RSW-result for the random cluster model $\lbrack 3 \rbrack$.