Universal finite-size scaling profiles on the torus

Let $d>d_{\mathrm{c},\alpha}$ and $s\in\mathbb{R}$. For each of the models self-avoiding walk, Ising, $|\varphi|^4$, percolation, and branched polymers, let $\chi_R(\beta)$ denote the torus susceptibility, $G_{R,\beta}(x)$ the torus two-point function, and $V=R^d$. Parametrize the critical window by

$$\beta_{\mathrm{c}}(s)=\beta_{\mathrm{c}}+s a_d V^{-\frac{2}{\gamma d_{\mathrm{c}}}},$$

where $\beta_{\mathrm{c}}$ is the infinite-volume critical point, $a_d>0$, and $f\sim g$ means $f/g\to1$. Then there is a model-dependent profile function $f:\mathbb{R}\to(0,\infty)$, common to the SR and LR versions of the model, and positive constants $a_d,b_d$, such that

$$\chi_R(\beta_{\mathrm{c}}(s))\sim b_d f(s)V^{\frac{2}{d_{\mathrm{c}}}},$$

and, for every $x\in\mathbb{T}_R^d$,

$$G_{R,\beta_{\mathrm{c}}(s)}(x)\sim G_{\beta_{\mathrm{c}}}(x)+\frac{b_df(s)}{V^{1-\frac{2}{d_{\mathrm{c}}}}}.$$

Universal-profile conjecture. The preceding susceptibility and two-point-function asymptotics hold for every listed model, with the same profile function for its SR and LR versions.

The conjecture gives precise universal finite-size scaling throughout the critical window, beyond the existence of a plateau. The surrounding text notes rigorous profile results on the hierarchical lattice and complete-graph analogues, but does not establish the stated lattice result.