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 \(a^{15/8}h\), 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.