13 problems

Let $T_n$ denote the number of steps needed for simple random walk in ${\mathbb{Z}}^2$ to completely cover the disc of radius $n$. Kesten and Révész established constants…

Let $\sigma_N$ be the cover time for $N$ searchers, let $\mathbb{E}[\sigma_1]$ be the mean cover time for a single searcher, and let $L$ and $D$ denote the characteristic length an…

Let $\alpha_c^1(\delta)$ be the critical exponent for exponential integrability of the cover time of a $\delta$-once-reinforced random walk, and let $X$ start from a specified vert…

Universal speed-up conjecture. There exist universal constants $C$ and $C'$ such that, for every graph $G$ and every $k$,

Let $C_N(4)$ be the cover time at deceleration parameter $\alpha=4$, let $N$ be the number of sites, let $g(0)$ be the Green function at the origin, and let $s$ be the Laplace-tran…

Let $C_N(2)$ be the cover time at deceleration parameter $\alpha=2$, let $N$ be the number of sites, let $g(0)$ be the Green function at the origin, and let $s$ be the Laplace-tran…

Consider dynamically accelerated or decelerated random walks whose cover-time statistics are related to the roughness of $1/f^{\alpha}$ Gaussian signals. Gaussian-to-exponential co…

Let $d\geq 3$ and let $C_N(\alpha)$ denote the cover time of a dynamically accelerated or decelerated random walk on an $N$-site lattice. Let $g(0)$ be the Green function for the c…

Let $C_N$ be the time taken by discrete- or continuous-time random walk to visit every vertex of the two-dimensional discrete torus graph $(\mathbb{Z}/N\mathbb{Z})^2$. Subleading-o…

Let $\tau_{\mathrm{cov}}$ be the cover time of a random walk on a two-dimensional box, and let $m_n$ denote the median of $\sqrt{\tau_{\mathrm{cov}}/n}$. Bramson–Zeitouni's tightne…

Let the cover time of a finite graph be the time required for a random walk to visit every vertex. For the two-dimensional discrete torus, consider the cover time under the suitabl…

Let $C_N$ denote the cover time of the discrete torus $(\mathbb{Z}/N\mathbb{Z})^d$ of side length $N$, where $d\geq 3$, and let $g(0)$ be the Green's function at the origin for sim…

Let $G_N=(\mathbb{Z}/N\mathbb{Z})^d$ be the discrete torus of dimension $d\geq 3$, let $C_N$ denote the cover time of $G_N$, and let $g(0)$ be the value at the origin of the Green'…