Critical three-dimensional Ising spin-field scaling limit

Let \(\langle\cdot\rangle_{\beta_c}\) be the infinite-volume zero-field Gibbs state at the critical inverse temperature \(\beta_c\) of the nearest-neighbor ferromagnetic Ising model on \(\mathbb Z^3\), where \(\beta_c\) is the infimum of the inverse temperatures for which the infinite-volume plus state has positive magnetization. The spins are \(\sigma_x\in\{-1,1\}\), and finite-volume weights are proportional to \(\exp(\beta\sum_{\langle x,y\rangle}\sigma_x\sigma_y)\), with the sum over nearest-neighbor edges. For \(a>0\), set \(e_1=(1,0,0)\), \(n_a=\lfloor a^{-1}\rfloor\), \(c_a=\langle\sigma_0\sigma_{n_ae_1}\rangle_{\beta_c}^{-1/2}\), and define a random tempered distribution by \[ \Phi_a(f)=a^3c_a\sum_{x\in\mathbb Z^3}\sigma_x f(ax),\qquad f\in\mathcal S(\mathbb R^3). \] As \(a\downarrow0\), \(\Phi_a\) converges in law in \(\mathcal S'(\mathbb R^3)\) to a non-Gaussian random distribution \(\Phi\). For every integer \(k\ge1\), its \(k\)-point correlation distribution has a smooth restriction \(S_k\) to pairwise distinct points, and there is \(\Delta_\sigma>0\) such that every conformal diffeomorphism \(\varphi:U\to V\) between open subsets of \(\mathbb R^3\) satisfies \[ S_k(\varphi(x_1),\ldots,\varphi(x_k))=\prod_{i=1}^k\lambda_\varphi(x_i)^{-\Delta_\sigma}S_k(x_1,\ldots,x_k),\qquad \lambda_\varphi(x)=|\det D\varphi(x)|^{1/3}, \] for all pairwise distinct \(x_1,\ldots,x_k\in U\).