313 problems

Let $G$) be a connected $d$-regular graph with adjacency matrix $A$ and $n$ vertices. Let $\xi_1,\xi_2,\ldots,\xi_n$ be orthogonal eigenvectors of $A$, with the conditions stated b…

Let $d\ge 1$. Suppose $D(x)$ has finite support $\mathcal{U}$. If $D(x)$ is uniformly distributed on $\mathcal{U}$, and if every $x\in \mathcal{U}$ is on the boundary of…

Let $(X_n)_{n\ge 0}$ be a centrally excited random walk on $\mathbb{Z}^d$, $d\ge 2$, which otherwise moves as simple random walk, and let $V_n=\{X_0,X_1,\ldots,X_n\}$. Kozma conjec…

Consider the parking process on $\mathbb{Z}^d$ in which each site initially contains a car with probability $p$ and a parking spot with probability $1-p$, independently, and cars p…

Let each site of $\mathbb{Z}^2$ be assigned an independent uniformly random initial direction among its four outgoing directions. Starting at the origin, the Eulerian walker turns…

Drift conjecture. The drift function satisfies

A high-dimensional random walk on a discrete point process is a random walk on such a point process in sufficiently large dimension. Failure conjecture. There are random walks on d…

Let a Galton–Watson tree be supercritical and condition it on survival. Consider the simple exclusion process on this tree in the constant speed model and in the variable speed mod…

Let $\Gamma<\mathrm{SL}_N(\mathbb{R})$ be a discrete subgroup, and let the random walk on $\Gamma$ have finite support. Its hitting measure is a measure on the boundary at infinity…

Let $G$ be a connected regular graph on $n$ vertices, and let $\tau$ denote the relaxation time of its random walk, defined by … where $\eta_2$ is the second-largest eigenvalue of…

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…

Let $G$ be a finite graph with vertex set $V$. For a starting vertex $v\in V$, write $\mathbb{E}_v$ for expectation for the simple random walk started at $v$, and let the…

Let $G$ be a virtually-$\mathbb{Z}$ group and let $H$ be a nontrivial finite group. Let $\lambda_c$ be the critical parameter such that the $\lambda$-stationary random walk on…

Singularity conjecture. If $\mathsf{m}$ has finite support, then the $\mathsf{m}$-stationary measure $\nu$ is singular to the Lebesgue measure class on $\operatorname{\mathcal{F}}$…

Let $G$ be a group, let $k$ satisfy $k \gg \log |G|$ and $\log k \ll \log |G|$, and let $\mathcal{G}_k$ be the Cayley graph of $G$ generated by $k$ independently and uniformly chos…

Let $G$ and $H$ be finite graphs that are $K$-roughly isometric, and suppose both have maximal degree at most $d$. Denote by $\tau_{\infty}(G)$ and $\tau_{\infty}(H)$ the…

Fix $n\in\mathbb{N}$. Let $d$ be the spatial dimension, let $\kappa\geq 0$ be the diffusion constant of the catalyst, and let $\kappa_p^{(n)}$ denote the critical values governing…

Let $R$ be a symmetric random walk on $\mathbb{Z}$ with step distribution satisfying … for some $0<\alpha<1$, and let $D_n$ denote the diameter of the aggregate after $n$ particles…

Tightness conjecture. The highest points of sufficiently long positive arches, and the lowest points of sufficiently long negative arches, are tight to the Brownian limit.

Let the Laplacian operator be the operator defined by the jump conditions in --, and let $h_1$ denote the function in Theorem. Uniqueness conjecture. The unique, up to multiplicati…

Let $n$ particles evolve according to the Attracting Random Walks (ARW) model on an arbitrary graph, with interaction parameter $\beta<0$. The mixing time is measured with respect…

The shell-based inpainting Gaussian-blur conjecture. If $u_0$ is independent of its $y$-coordinate, then

Let $(T_n(k))_{k \geq 0}$ denote the flip walk on triangulations of the sphere with $n$ vertices, and let its mixing time be measured with respect to the uniform distribution on su…

Let ${\cal LF}_n$ be a locally free group, and consider its random walk with word length $K$ and drift $l$, the limiting expected normalized word length. Let $T(w)$ be the roof of…

Let ${\cal LF}_n^+$ be the locally free semigroup (the heap) and ${\cal LF}_n$ the locally free group (the colored heap). For a random walk word $w_N$, let $T(w_N)$ denote its roof…