The non-Gaussian limit and universal profile conjecture for the high-dimensional Ising model

Let $d\geq 4$, let $\mathbb{T}_r$ be the $d$-dimensional torus of side length $r$, and let $S_r$ be the average field. For $s\in\mathbb{R}$, set $\beta=\beta_c+s w_r$, where

$$w_r=\begin{cases}a_4(\log r)^{-1/6}r^{-2}&(d=4),\\a_dr^{-d/2}&(d>4),\end{cases}\qquad h_r=\begin{cases}b_4(\log r)^{1/4}r^{-1}&(d=4),\\b_dr^{-d/4}&(d>4),\end{cases}$$

with suitably chosen constants $a_d,b_d>0$. Let $f(s)=\int_{\mathbb{R}}x^2\,\mathrm d\sigma_s$, where

$$\mathrm d\sigma_s=\frac{e^{-x^4/4+sx^2/2}\,\mathrm dx}{\int_{\mathbb{R}}e^{-x^4/4+sx^2/2}\,\mathrm dx}.$$

Non-gaussian limit and universal profile conjecture. As $r\to\infty$, $h_r^{-1}S_r$ converges in distribution under $\langle\cdot\rangle_{\beta_c+sw_r}^{\mathbb{T}_r}$ to $\mathrm d\sigma_s$, with convergence of all moments. Moreover,

$$\chi^{\mathbb{T}_r}(\beta_c+sw_r)\sim r^dh_r^2f(s)=b_d^2f(s)\begin{cases}(\log r)^{1/2}r^2&(d=4),\\r^{d/2}&(d>4),\end{cases}$$ $$\lim_{r\to\infty}g^{\mathbb{T}_r}(\beta_c+sw_r)=3-\frac{\int_{\mathbb{R}}x^4\,\mathrm d\sigma_s}{\left(\int_{\mathbb{R}}x^2\,\mathrm d\sigma_s\right)^2},$$

and, for $x\in\mathbb{T}_r$,

$$\tau^{\mathbb{T}_r}_{\beta_c+sw_r}(0,x)-\tau_{\beta_c}(0,x)\sim h_r^2f(s)=b_d^2f(s)\begin{cases}(\log r)^{1/2}r^{-2}&(d=4),\\r^{-d/2}&(d>4).\end{cases}$$

These predictions give a universal non-Gaussian finite-size scaling profile for the average field, susceptibility, plateau, and renormalised coupling constant. They are conjectured by universality from rigorous results for hierarchical $|\varphi|^4$ models, but remain unproved for the nearest-neighbour Ising model.