Mass Scaling of the Near-Critical 2D Ising Model Using Random Currents

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 \(h \rightarrow 0\). This was previously proven with CLE-methods in Camia et al. in (Commun Pure Appl Math 73(7):1371–405, 2020). 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 (Aizenman et al. in Invent Math 216:661–743, 2018) as well as a near-critical RSW-result for the random cluster model (Duminil-Copin and Manolescu in Planar Random-Cluster Model: Scaling Relations, 2020).