Anisotropic KPZ conjecture for two-dimensional interface growth

Let $v=v(\rho)$ be the slope-dependent asymptotic growth speed for a two-dimensional stochastic interface, with local average slope $\rho\in\mathbb R^2$. Let $\alpha$ and $\beta$ denote the roughness and growth exponents, and let the Edwards–Wilkinson exponents be

$$\alpha_{EW}(2)=0,\qquad \beta_{EW}(2)=0.$$

Anisotropic KPZ conjecture. If the Hessian matrix $D^2v(\rho)$ has two eigenvalues of the same sign, equivalently if $\det(D^2v(\rho))>0$, then

$$\alpha\ne\alpha_{EW}(2)=0,\qquad \beta\ne\beta_{EW}=0.$$

If instead $\det(D^2v(\rho))\leq 0$, then

$$\alpha=\alpha_{EW}(2)=0,\qquad \beta=\beta_{EW}=0.$$

This conjecture predicts that the sign structure of the Hessian of the growth speed distinguishes genuinely anisotropic KPZ behavior from Edwards–Wilkinson behavior in two dimensions. The statement is presented as an expectation for the long-time fluctuation exponents of the stochastic growth models considered in the paper.