9 problems

Erdős–Taylor conjecture. For $d=2$, the maximal local time satisfies

Let $(x_n)_{n\geq 0}$ be the planar random walk defined by $x_0=0$, with $x_{n+1}$ chosen uniformly on the unit circle centered at $x_n$, conditioned so that the segment…

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…

Optimal-exponent conjecture. The optimal exponent is

Let $\mu$ be a translation-invariant, ergodic percolation measure on $\mathbb{Z}^2$ that almost surely has infinitely many infinite clusters. Let $\mu_{\operatorname{hp}}$ denote i…

Let $k \in \mathbb{Z}_{\geq 1}$. Let $\mu$ be a translation-invariant, ergodic, finite-energy, $k$-dependent edge percolation measure on $\mathbb{Z}^2$. Full-plane trichotomy conje…

Let $(A_n)_{n\geq 0}$ be the DLA cluster process in the wedge $W_{\theta_1,\theta_2}$, and define its growth rate by … where $\operatorname{diam}$ denotes Euclidean diameter, and l…

Let $A_\infty$ be the limiting diffusion-limited aggregation cluster in a wedge, and interpret its arms as the ends of the graph $A_\infty$. One-arm conjecture. There exists…

Let $\Omega$ be a simply connected open domain in the plane, and let $A,B,C,D$ be four boundary points of $\Omega$ in clockwise order. For critical percolation on the $1/n$-scaled…