8 problems

Quasi-loop theorem conjecture. Theorem 4.6 should hold for any Euclidean net. Extending the result would give quasi-loop control for loop-erased random walks on all Euclidean nets,…

Four-dimensional logarithmic-correction conjecture. The accurate upper-bound scale for the length of $L$ is

Critical-dimension conjecture. The critical dimension is

Loop-erased random walk lower-bound conjecture. The expected length of the loop-erasure of $R$ satisfies

Uniqueness conjecture. If, for each $i\in\{1,2\}$, the union of $\mathrm{K}_i$ and all loops from $\mathrm{BS}$ intersecting $\mathrm{K}_i$ has the same law as the trace of Brownia…

Let $z\in\mathbb{D}$, let $\epsilon>0$ be sufficiently small, and let $Z$, $W^w$, $\xi_x$, and $\sigma_{\mathbb{D}}$ have the meanings used in the loop-erased-random-walk estimates…

Let $D$ be a bounded, simply connected domain containing the origin whose boundary is a Jordan curve, and let $U,V$ be disjoint, closed subarcs of $\partial D$. For the lattice app…

Let $x \in \mathbb{Z}^2$ be a neighbor of the origin, and let $p_N$ be the probability that $x$ lies on the loop-erased random walk from $o$ to $\partial G_N$. Loop-erased random w…