Mass scaling of the near-critical Ising model in dimensions $d\geq 4$

Abstract

We study the Ising model on $\mathbb{Z}^d$ with $d\geq 4$ and derive near-critical bounds on the truncated two-point function $\langle\sigma_0;\sigma_x\rangle_{\beta,h} := \langle\sigma_0\sigma_x\rangle_{\beta,h} - \langle\sigma_0\rangle_{\beta,h}\langle\sigma_x\rangle_{\beta,h}$ at parameters $\beta\leq\beta_c$ and $h\geq 0$. As a corollary, we obtain that the associated mass (or exponential decay rate) is equal to \begin{equation*} \max\bigl((\beta_c-\beta)^{1/2},h^{1/3}\bigr)^{1+o(1)}, \end{equation*} where $o(1)$ tends to $0$ as $(\beta,h)$ tends to $(\beta_c,0)$. The proof combines the corresponding result at $h=0$, recently established by Duminil-Copin and Panis, with an interpolation argument inspired by Aizenman and Fern\'andez and carried out via the random current representation of the model.