Stationary distributions in higher dimensions

Definitions and mathematical statement

Consider exclusion on $\mathbb Z^2$: a particle at $x$ attempts jumps to $x+e_i$ at rate $p_i$ and to $x-e_i$ at rate $q_i$, for $i=1,2$, and a jump is suppressed when its destination is occupied. Assume $p_i>q_i>0$ and $p_1+q_1+p_2+q_2=1$. Write $m=(p_1-q_1,p_2-q_2)$ for the mean jump vector.

Fix a nonzero integer vector $v\in\mathbb Z^2$ with $v\cdot m>0$. Does there exist a probability measure $\mu$ on $\{0,1\}^{\mathbb Z^2}$ which is stationary for this process, is invariant under every lattice translation $z$ satisfying $z\cdot v=0$, and obeys \[ \lim_{n\to-\infty}\mu(\eta(nv)=1)=0, \qquad \lim_{n\to+\infty}\mu(\eta(nv)=1)=1? \] Here stationarity means that the law remains $\mu$ at every nonnegative time when the initial configuration has law $\mu$. Translation invariance is only required along the interface, in directions perpendicular to $v$; no independence between occupation variables is assumed.

The integer-vector convention spells out the rational-direction formulation explicitly mentioned in the source: one clears denominators before evaluating the configuration at $nv$. The strict scalar-product inequality is equivalent to the angle between the proposed density-gradient direction and the drift being less than $\pi/2$. The desired conclusion is a family of stationary measures for every admissible direction and every irreducible nearest-neighbor kernel, not merely for a direction parallel to a coordinate axis or a direction permitting an explicit product measure.

Short English statement

Can a stationary two-dimensional exclusion interface point in every rational direction having positive projection onto the particle drift?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 9600002 (AMR-095-0002), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Liggett - Some Open Problems (2012), Problem 2, PDF page 2. Original source indexed by AMR-095-0002. https://web.archive.org/web/20150919235903id_/http://www.math.ucla.edu/~tml/open.pdf.