A simple proof of exponential decay for the near-critical planar Ising model

For the Ising model defined on $a{\mathbb{Z}}^2$ at critical temperature with external field $a^{15\mathrm{/}8}h$, we give a simple and elementary proof that its truncated two-point function decays exponentially. The proof combines the high temperature expansion, random-cluster and random current representations. A new input in the proof is that, in the near-critical sourceless single current measure, there are many loops formed by a path on $a{\mathbb{Z}}^2$ with diameter of order $1$, together with two external edges that connect the path’s endpoints to the ghost.