Pseudocritical finite-size scaling for free boundary conditions

Let $\Lambda_{R,d}$ be a discrete $d$-dimensional box of side length $R$, with volume $V=R^d$, and let superscript $\mathrm{F}$ denote free boundary conditions. For the models self-avoiding walk, Ising, $|\varphi|^4$, percolation, and branched polymers, let $\chi_R^{\mathrm{F}}$ and $G_{R,\beta}^{\mathrm{F}}(0,x)$ denote the free-boundary susceptibility and two-point function. Let $\beta_{\mathrm{c}}$ be the infinite-volume critical point and $\nu$ the correlation-length exponent.

Pseudocritical-profile conjecture. For $d>d_{\mathrm{c},\alpha}$, there is a pseudocritical point

$$\beta_{R,\mathrm{c}}=\beta_{\mathrm{c}}+\operatorname{const}R^{-1/\nu}$$

shifted into the ordered phase. With

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

there are positive constants $a_d,b_d$ such that, as $R\to\infty$,

$$\chi_R^{\mathrm{F}}(\beta_{R,\mathrm{c}}(s))\sim b_df(s)V^{\frac{2}{d_{\mathrm{c}}}},$$

where $f$ is the same profile function from Table~, and, for $x\in\Lambda_{R,d}$ with $|x|/R\to0$,

$$G_{R,\beta_{R,\mathrm{c}}(s)}^{\mathrm{F}}(0,x)\sim G_{\beta_{\mathrm{c}}}(x)+\frac{b_df(s)}{V^{1-\frac{2}{d_{\mathrm{c}}}}}.$$

The conjecture asserts that the periodic-boundary scaling window, susceptibility scale, plateau scale, and universal profiles are exactly reproduced at a shifted free-boundary pseudocritical point. The source notes that the free-boundary statement is proved only in some cases and that the constants differ from those for periodic boundary conditions.