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 $h\to 0$. This was previously proven with CLE-methods in $[1]$. 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 $[2]$ as well as a near-critical RSW-result for the random cluster model $[3]$.

16 pages, 5 figures