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

Abstract

We study the Ising model on ${\mathbb{Z}}^d$ with $d\ge 4$ and derive near-critical bounds on the truncated two-point function $⟨{\sigma }_0;{\sigma }_x{⟩}_{\beta ,h}:=⟨{\sigma }_0{\sigma }_x{⟩}_{\beta ,h}-⟨{\sigma }_0{⟩}_{\beta ,h}⟨{\sigma }_x{⟩}_{\beta ,h}$ at parameters $\beta \le {\beta }_c$ and $h\ge 0$. 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 $(\beta ,h)$ 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 Fernández and carried out via the random current representation of the model.

32 pages