8 problems

Martin boundary conjecture. Almost surely, $\partial \mathbb{D}$ is a realization of the Martin boundary of $G$.

Let ${\cal Z}$ be the graded graph of zigzag diagrams, with level-${n}$ vertices ${\cal Z}_n$. For a sequence $(\lambda_n\in{\cal Z}_n)$, let $\iota_n(\lambda_n)$ be its image in t…

Homeomorphism conjecture. In case (B2), the Martin boundary is homeomorphic to $\overline{\mathcal W}_0$.

Vanishing-kernel conjecture. If $\frak{b}=+$ and $f(R_c)=+$, then

Let $K$ be the transition kernel in the setting above, let $x_n\to b\in\hat{E}$, and write $\pi_b(\cdot)=\lim_{x_n\to b}\chi(x_n,\cdot)$. Suppose the Jacka–Roberts Condition [5] ho…

Let $\mathcal{Z}$ be the zigzag graph, let $\partial_{M}\mathcal{Z}$ and $\partial_{\min}\mathcal{Z}$ denote its Martin and minimal entrance boundaries, and let $\mathcal{U}^{(2)}$…

Let $G_R$ denote the Green's function at the spectral radius $r=R$ for the random walk under consideration, and let the Martin boundary be the boundary associated with this Green's…

Stability conjecture. The $t$-Martin compactification should be stable in general.