15 problems

Uniqueness conjecture. If

For $d\geq3$, integers $n\geq2$ and $m\geq3$, define … where $\|x_1-x_2\|$ is graph distance. Separation-distance conjecture. (1) There exist $C(d,m),c(d,m)>0$ such that, for all…

For $d\geq3$ and $n\geq2$, let $\mathfs{M}_n(\mathbb{Z}^d)$ be the least positive harmonic measure among subsets of $\mathbb{Z}^d$ of cardinality $n$. Two lattice vertices are $*$-…

Let $\psi(\mathbb{Z}^2)$ denote the reciprocal extremal factor governing the exponential lower bound for positive harmonic measures on finite subsets of $\mathbb{Z}^2$. The propose…

Let $G$ be the group of isometries of the hyperbolic plane $\mathbb{H}^2$. Let $\mu$ be a finitely supported probability on $G$ whose support generates, as a semigroup, a discrete…

Let $J$ be a conformal self-similar set of dimension $\delta<1$, and let $\omega_{\mathbb C\setminus J}$ denote harmonic measure for its complement. Dimension-drop conjecture. The…

Let $\Omega\subset\mathbb R^n$ be a one-sided NTA domain whose Green function with some pole $p$ satisfies, for some $\delta>0$ and constants $c_1,c_2>0$, bounds of the form … for…

Square-spiral conjecture. Asymptotically, the square spiral realizes the least positive value of harmonic measure, in the sense that

Let $\widehat{q}(y)$ be the quantity defined in the paper from the harmonic measure of the embedded graph, and let $y$ tend to infinity in the Euclidean embedding. Harmonic-measure…

Let $D$ be a simply connected domain containing the origin, and let $\delta$ be the lattice spacing of the square lattice. Let $\mu_\delta(0,|dz|;D)$ be the discrete exit distribut…

Let $\mathscr{V}^\lambda$ be the hyperbolic Poisson–Voronoi graph, and let $\nu_0$ be the harmonic measure on $S^1$ obtained from the almost-sure limit of simple random walk on…

Let $G$ be an infinite graph with the doubling property: there exists a universal constant $C>0$ such that for every $r>0$ and every vertex $x$, … where $B(x,r)$ is the ball of rad…

Let $G$ be an open subset, with components whose harmonic measures are considered. A positive measure $\mu$ on $G$ is a Carleson measure when it satisfies the relevant Carleson emb…

Let $G$ be a domain conformally equivalent to a circular domain $W$, and let $\phi:G\to W$ be a conformal map. A positive measure on $G$ is a Carleson measure if it satisfies the C…

Let $K$ be a domain, let $x^0$ be a point, and let $\beta$ be a positive measure. A positive harmonic ball is a set of the form $D(x^0,\alpha)$ satisfying the defining harmonic-bal…