Monotonicity along increasing curves in the antiferromagnetic isotropic Ashkin–Teller model

Let $\gamma:(0,1)\to\mathbb{R}^3$ be a curve in the antiferromagnetic FKG regime, with all components strictly increasing. For parameters $(J_i,J'_i,U_i)$, let $\langle\tau_0\rangle_{\mathsf{B}_n,J_i,J'_i,U_i}^{+,\mathrm{f}}$ denote the corresponding finite-volume expectation, and let $\mathcal{B}_r(x)$ denote the three-dimensional Euclidean ball of radius $r>0$ centred at $x\in\mathbb{R}^3$.

Monotonicity conjecture. For $\beta_1,\beta_2\in(0,1)$ with $\beta_1<\beta_2$, there exists $\varepsilon>0$ such that

$$\langle\tau_0\rangle_{\mathsf{B}_n,J_1,J'_1,U_1}^{+,\mathrm{f}}\leq\langle\tau_0\rangle_{\mathsf{B}_n,J_2,J'_2,U_2}^{+,\mathrm{f}}$$

whenever $(J_i,J'_i,U_i)\in\mathcal{B}_\varepsilon(\gamma(\beta_i))$ for $i=1,2$.

This property, together with the paper’s sharpness theorem, would imply sharpness along all strictly increasing curves in the isotropic phase diagram when $d=2$. The source also explains that the transition should occur on the self-dual curve $\sinh 2J=e^{-2U}$; the conjectured monotonicity itself remains open.