Critical-window scaling conjecture for the weakly coupled $|\varphi|^4$ model

Let $\operatorname{\bf NG}_{s}$ be the probability measure whose moment generating function is specified by

$$\operatorname{\bf NG}_s[e^{(\psi,f)}]\propto\int_{\mathbb R}e^{(\psi,\Phi(f))}e^{-\frac14|y|^4-s|y|^2}\,\mathrm{d}y.$$

Under the assumptions of Theorem~, let $w_N$ denote the critical-window scale, and let $\varphi$, $f_N$, ${\sf b}_N$, $\psi$, $f$, and $c_3$ be as in the theorem. Critical-window scaling conjecture. There exists $c>0$ such that

$$\lim_{N\to\infty}\left\langle e^{(\varphi,f_N)/{\sf b}_N}\right\rangle_{g,\nu_c+sw_N,N}=\operatorname{\bf NG}_{cs}\left(e^{(\psi,f)/c_3^{1/4}}\right).$$

This would strengthen the predicted critical-window susceptibility scaling profile by identifying the full limiting moment-generating function. The conjecture is expected to follow by differentiating the renormalisation-group dynamics; a simpler version is known in the hierarchical setting.