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

Source 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 $\langleσ_0;σ_x\rangle_{β,h} := \langleσ_0σ_x\rangle_{β,h} - \langleσ_0\rangle_{β,h}\langleσ_x\rangle_{β,h}$ at parameters $β\leqβ_c$ and $h\geq 0$. As a corollary, we obtain that the associated mass (or exponential decay rate) is equal to \begin{equation*} \max\bigl((β_c-β)^{1/2},h^{1/3}\bigr)^{1+o(1)}, \end{equation*} where $o(1)$ tends to $0$ as $(β,h)$ tends to $(β_c,0)$. The proof combines the corresponding result at $h=0$, recently established by Duminil-Copin and Panis, with an interpolation argument inspired by Aizenman and Fernández and carried out via the random current representation of the model.