The percolation critical scaling profile conjecture

Consider bond percolation on the complete graph with $V$ vertices, with edge probability $p/V$. Let $\tau_{V,01}(p)=\mathbb{P}_{p/V}(0\leftrightarrow1)$ be the two-point function and let $\chi_V^{\rm perc}(p)=1+(V-1)\tau_{V,01}(p)$ be the susceptibility. Let $W^*$ be a Brownian excursion of length $1$, define its area moment-generating function by

$$\Psi(x)=\mathbb{E}\exp\left[x\int_0^1W^*(t)\,\mathrm{d}t\right],$$

and, for $s\in\mathbb{R}$, define

$$f_{\rm perc}(s)=\int_0^\infty x^2\,\mathrm{d}\sigma_s,$$

where

$$\mathrm{d}\sigma_s=\frac{1}{\sqrt{2\pi}}x^{-5/2}\Psi(x^{3/2})\exp\left(-\frac16x^3+\frac{s}{2}x^2-\frac{s^2}{2}x\right)\,\mathrm{d}x.$$

Percolation scaling-profile conjecture. There exist constants $a,b>0$ such that, as $V\to\infty$,

$$\tau_{V,01}(1+sV^{-1/3})\sim bV^{-2/3}f_{\rm perc}(as),$$ $$\chi_V^{\rm perc}(1+sV^{-1/3})\sim bV^{1/3}f_{\rm perc}(as).$$

This conjecture predicts the analogue of the critical scaling profile theorem for percolation on the complete graph. The powers $V^{-1/3}$, $V^{-2/3}$, and $V^{1/3}$ are proved for the corresponding torus result in dimensions $d>6$, but the role of the profile $f_{\rm perc}$ was previously conjectured and is therefore not established here.