22 problems

Mauldin–Monticino–Weizsäcker's transience conjecture. In dimension $3$, this scheme is always transient, in the sense that every bounded set is visited only finitely often almost s…

Let an RWCE be a random walk in a changing environment, and call it non-adaptive when the environment's evolution is independent of the walk's history. The source discusses recurre…

Bertoin's recurrence conjecture. The random walk $\bm{S}$ is recurrent. The theorem establishes recurrence in this regime under the stronger moment assumption…

Let $d\in\{1,2\}$, let $g$ be a kernel satisfying … and let $\rho$ be a profile function satisfying the source's condition for some $\delta\geq 2$. Define … Assume that … Let…

For $d=2$, let $\rho$ be the profile function and let $\beta$ and $\theta$ be the parameters represented by the phase diagram in part (b). The transient regime is the region in whi…

Let $\mathbb{G}=(V,E)$ be a graph with time-dependent, symmetric non-negative edge conductances $\{\Pi_t\}$, and suppose the conductances are independent of the random walk. Assume…

Let $\mathbb{G}_\infty$ be a graph, and let $\{\mathbb{G}_t\}_{t\in\mathbb{N}}$ be a graph sequence dynamically growing towards $\mathbb{G}_\infty$. Let $\{X_t\}_{t\in\mathbb{N}}$…

Let $G=(V,E)$ be a vertex-transitive graph with distinguished vertex $o$. At each vertex $x\in V\setminus\{o\}$, let the frog number $\eta(x)$ be an i.i.d. random variable, let exa…

Ergodic-environment conjecture. Theorem should extend to ergodic cookie environments, possibly under some additional ellipticity assumptions; that is,

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 $\mathbb{D}_0\subseteq\mathbb{Z}^d$, $d\ge3$, be the initial domain, and let the once boundary-triggered interaction (OBT interaction) evolve the domain as a simple random walk…

A once reinforced random walk is a nearest-neighbor walk whose transition probabilities are modified according to the strength of reinforcement applied to previously traversed edge…

Let $\bmathbb{G}_\infty$ be a graph with non-decreasing edge conductances $c_t(e)$, converging to $c_\infty(e)$, and consider the corresponding time-dependent random walk…

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…

Moving to $\bmathbb{Z}^d$, $d\bge3$, consider once reinforced random walk, in which the transition rule is modified according to the strength of the reinforcement. Last author's se…

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…

A two-dimensional random walk on a discrete point process is the random walk considered in the paper when the ambient dimension is $2$. Existence conjecture. There are transient tw…