The effective critical point conjecture for free-boundary high-dimensional Ising models

Let $\Lambda_r$ be the box of side length $r$ and consider the Ising model with free boundary conditions in dimensions $d\geq 4$. Let $\beta_c$ denote the infinite-volume critical point. An effective critical point is a value $\beta_{c,r}$ satisfying

\beta_{c,r}=\beta_c+v_r,\qquad v_r\asymp\begin{cases}(\log r)^{1/3}r^{-2}&(d=4),\r^{-2}&(d>4). \end{cases}

Free-boundary effective critical point conjecture. There exists such a $\beta_{c,r}$ for which all statements of the non-Gaussian limit and universal profile conjecture hold with $\beta_c$ replaced by $\beta_{c,r}$; in particular, the plateau phenomenon occurs for free boundary conditions at $\beta_{c,r}$. This conjecture transfers the torus finite-size scaling predictions to free boundary conditions, with a dimension-dependent shift of the critical point. Its resolution is not supplied in the source.