Invariant-probability dichotomy for planar Dirichlet environments

Let $X$ be a random walk in Dirichlet environment on ${\mathbb Z}^2$ with positive parameters $(\alpha_1,\ldots,\alpha_4)$. Define

$$\kappa=2\sum_{i=1}^4\alpha_i-\min(\alpha_1+\alpha_3,\alpha_2+\alpha_4),\qquad d_\alpha=\sum_{i=1}^4\alpha_i e_i.$$

A local accelerating function $\gamma$ is a positive function depending only on the environment in some finite box; it changes the jump rate from $x$ to $y$ to $\gamma(\tau_x\omega)\omega(x,y)$. Invariant-probability dichotomy conjecture. If $\kappa>1$, there exists a unique invariant probability from the point of view of the particle if and only if $d_\alpha\neq0$. If $\kappa\leq1$, there exists no invariant probability from the point of view of the particle; however, there exists a local accelerating function $\gamma$ for which the accelerated particle has a unique invariant probability if and only if $d_\alpha\neq0$.

The conjecture synthesizes the paper's results on recurrence, transience, traps, and acceleration. Its resolution depends on the expected equivalence between directional transience and condition $\mathbf{(T')}$, as well as on the zero-drift recurrence conjecture.