Critical clustering for linearly interacting diffusions

Let $D\subset\mathbb R^d$ be open, bounded, and convex, let ${\cal W}$ be a renormalization class on $\overline D$, and suppose that the asymptotic fixed-point equation in the zero-ratio case has a unique solution $w^\ast\in{\cal W}$. Let $\sigma$ be a continuous root of a diffusion matrix $w\in{\cal W}$. Let $\mathbf x=(\mathbf x_\xi)_{\xi\in\mathbb Z^2}$ be a $\overline D^{\mathbb Z^2}$-valued solution of

$$\mathrm d\mathbf x_\xi(t)=\sum_{\eta:\,|\eta-\xi|=1}\big(\mathbf x_\eta(t)-\mathbf x_\xi(t)\big)\,\mathrm dt+\sigma(\mathbf x_\xi(t))\,\mathrm dB_\xi(t),$$

with $\mathbf x_\xi(0)=\theta\in\overline D$ for all $\xi\in\mathbb Z^2$. Critical clustering. Then

$$\mathbf x_\xi(t)\underset{t\to\infty}{\Longrightarrow}I^\theta_\infty\qquad(\xi\in\mathbb Z^2),$$

where $(I^\theta_s)_{s\geq0}$ is the diffusion with generator $\sum_{i,j}w^\ast_{ij}(y)\frac{\partial^2}{\partial y_i\partial y_j}$ and initial condition $I^\theta_0=\theta$. This is the proposed clustering law for symmetric nearest-neighbor interaction on the critical two-dimensional lattice. The source presents it as a conjectural consequence of the renormalization limits and states no resolution; the broader critical-clustering program is described as open.