9 problems

A two-dimensional random walk on a discrete point process is a random walk in dimension $2$ whose state space is a discrete point process. Existence conjecture. There are transient…

A nonadaptive random walk in changing environment (nonadaptive RWCE) is a random walk whose edge weights are chosen in advance and therefore do not depend on the walker's location.…

Consider a labyrinth evolving by adding edges between adjacent vertices of $\mathbb{Z}^2$, without removing existing edges, and let Theseus perform the random walk on the resulting…

Let $(p_j)_{j\in\mathbb Z}$ be the transition-probability parameters of the anisotropic walk, with $p_j>0$. The walk is called transient if it visits each state only finitely often…

Consider a probing simple random walk (PSRW) $\{K_t\}$ on growing domains $\{\mathbb{D}_t\}$ in $\mathbb{Z}^d$, $d\ge2$, starting at $K_0=0$ with $\mathbb{D}_0=\{0\}$. At time $t$,…

Let $\bmathbb{G}_\infty$ be a fixed graph of uniformly bounded degrees on which simple random walk is recurrent, and let $\bmathbb{G}_t$ be non-decreasing subgraphs with…

Let $\bmathbb{G}_t$ and $\bmathbb{G}'_t$ be graphs non-decreasing in $t$, of uniformly bounded degrees, with $\bmathbb{G}_t\subseteq\bmathbb{G}'_t$ for every $t$. Let…

Let $d\ge3$, and let $\bmathbb{D}_t\subseteq\mathbb{Z}^d$ be connected and non-decreasing, with…

Non-necessity conjecture. This condition, which is sufficient for recurrence of the two-dimensional random walk, is not necessary. The claim would show that recurrence can hold wit…