25 problems

Schramm's conjecture. LERW has a conformally invariant scaling limit characterized by Schramm-Loewner evolution with parameter $2$.

Let $\hat{S}$ be the loop-erased random walk in two dimensions, and let … Define the growth exponent by ${\bf E}[\hat{\sigma}_n]\approx n^\alpha$, or, when the limit is not known t…

Let $G_n$ be a sequence of finite graphs satisfying either the conditions of Case 1 stated above or $G_n=\mathbb{Z}_n^4$. Consider the length of the loop-erased random walk on…

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,…

The paper defines isotropic graphs as a class of graphs intended to preserve the conformal features relevant to scaling limits of random walk and loop-erased random walk. It also d…

Let a loop-erased random walk in three dimensions be run until it reaches distance $r$, and let its number of steps be the quantity under discussion. Growth exponent conjecture. Ph…

Fomin's conjecture. The statement of Fomin's identity extends to nondiscrete Markov processes by passing to a limit in discrete approximations; the same procedure establishes the w…

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

Let $K$ be a scaling limit of loop-erased random walk on $\mathbb{Z}^3$, and let the loop-erased property be the property stated in the referenced conditional uniqueness theorem. S…

For $kappa=2$, let $\lambda$ be the cross-ratio and let … The function $U_2(\lambda)$ is conjectured to give the $kappa=2$ counterpart of $G^{(12)(34)}$, while identifying…

Let $g$ denote the one-point function associated with the scaling limit of loop-erased random walk on the unit domain, as introduced in the surrounding discussion. Differentiabilit…

Six-body-term removal conjecture. The modified action obtained from $S$ by removing the six-body term $S_{6}$ should lead to the same behaviour as the original action $S$.

Let a two-dimensional loop-erased random walk (LERW) be obtained by chronologically erasing loops from a random walk on a two-dimensional lattice. Its growth exponent describes the…

Let $D$ be a domain, let ${\cal K}$ denote the scaling limit of loop-erased random walk in $D$, and let $Y$ be the union of ${\cal K}$ and the loops from an independent Brownian lo…

LERW bottleneck exponent conjecture. By analogy with $\mathrm{SLE}_2$, this estimate should hold with $(r/R)^{3/2}$ in place of $(r/R)$, where

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…

Three-dimensional growth exponent conjecture. Physicists conjecture that there exists $\alpha$ such that

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 $Y_n$ be the time-reversed loop-erased random walk in the unit disk, let $\nu_{Y_n}$ be its occupation measure, let $M_n$ denote the corresponding loop-erased-walk length quant…

Let $D\subseteq\mathbb{C}$ be simply connected with $0\in D$, let $\phi:D\to\mathbb{D}$ be the unique conformal map with $\phi(0)=0$ and $\phi'(0)>0$, and let $D^n$ be the…

Let $\mathbb{D}$ be a simply-connected open domain of $\mathbb{C}$ other than $\mathbb{C}$. Couple a Brownian motion on $\mathbb{D}$, killed when it hits $\partial\mathbb{D}$, with…

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…