Scaling-limit factorization conjecture for the non-integrable Ising model

Let $U$ be the interaction appearing in the Hamiltonian, let $\beta^*$ be an inverse temperature, and let $\beta^*(\lambda)$ and $x^*=e^{-2\beta^*J}$ be as below. For a finite domain $\Omega\subset\mathcal{V}^{\scriptscriptstyle{\color{gray}\bullet}}$, let $\xi=(h_1,h_2,\ldots,h_{2N})$ be a collection of vertices in the relevant clustered-vertex set, and let $P_o$ be associated external boundary conditions. Let $x_c=\sqrt{2}-1$, and let $\langle\cdot\rangle_{\Omega,x_c}^{+}$ denote the critical Ising Berezin measure. There exists a collection of real numbers $\{\zeta_{\Omega,\lambda}^{P_o}(h)\mid h\in\mathscr{V}_C\Omega\}$ and a remainder $R_{\Omega,x^*,\beta^*,\lambda}^{P_o}(\xi)$ such that

Scaling-limit factorization conjecture. There exists $\lambda_0=\lambda_0(U)>0$ such that, for all $|\lambda|\leq\lambda_0$, there exists $\beta^*=\beta^*(\lambda)$ for which, for every such $\Omega$, $\xi$, and $P_o$,

$$\big\langle \varphi_{h_1}\varphi_{h_2}\cdots\varphi_{h_{2N}}\big\rangle_{\Omega,x^*,\beta^*,\lambda}^{P_o}=\zeta_{\Omega,\lambda}^{P_o}(h_1)\cdots\zeta_{\Omega,\lambda}^{P_o}(h_{2N})\big\langle \varphi_{h_1}\varphi_{h_2}\cdots\varphi_{h_{2N}}\big\rangle_{\Omega,x_c}^{+}+R_{\Omega,x^*,\beta^*,\lambda}^{P_o}(\xi).$$

The factors $\zeta_{\Omega,\lambda}^{P_o}$ are uniformly bounded, and there are constants $\theta>0$ and $\zeta_{\mathrm{bulk},\lambda}\in\mathbb{R}$ such that

$$\zeta_{\Omega,\lambda}^{P_o}(h)=\zeta_{\mathrm{bulk},\lambda}+O\big((\operatorname{dist}(h,\partial\Omega))^{-\theta}\big)$$

uniformly on $\Omega$ and $P_o$ for sufficiently large distance to the boundary. For two different domains $\Omega$ and $\Omega'$ with boundary conditions $P_o$ and $P_o'$, respectively,

$$\zeta_{\Omega,\lambda}^{P_o}(h)-\zeta_{\Omega',\lambda}^{P_o'}(h)=O\big((\operatorname{dist}(h,\Omega\triangle\Omega'))^{-\theta}\big)+O\big((\operatorname{dist}(h,P_o\triangle P_o'))^{-\theta}\big),$$

and

$$R_{\Omega,x^*,\beta^*,\lambda}^{P_o}(\xi)=O\left(\min_{1\leq i\neq j\leq 2N}|h_i-h_j|^{-1-\theta}\right)$$

uniformly on $\Omega$ and $P_o$. The conjecture proposes that the interacting non-integrable model has the same scaling-limit correlations as the critical isotropic square-lattice Ising model, up to local multiplicative renormalization factors and a controlled remainder. Establishing this would yield locally uniform convergence of the martingale observable to the planar Ising scaling limit.