Related papers: Sharp one-point estimates and Minkowski content fo…

In this work, we consider loop-erased random walk (LERW) in three dimensions and give an asymptotic estimate on the one-point function for LERW and the non-intersection probability of LERW and simple random walk in three dimensions for…

Let $M_{n}$ be the length (number of steps) of the loop-erasure of a simple random walk up to the first exit from a ball of radius $n$ centered at its starting point. It is shown in [18] that there exists $\beta \in (1, \frac{5}{3}]$ such…

Let $\beta$ be the growth exponent of the loop-erased random walk (LERW) in three dimensions. We prove that the scaling limit of 3D LERW is $h$-H\"older continuous almost surely for all $h < 1/\beta$, while not $1/\beta$-H\"older continuous…

We study the capacity of loop-erased random walk (LERW) on $\mathbb{Z}^d$. For $d\geq4$, we prove a strong law of large numbers and give explicit expressions for the limit in terms of the non-intersection probabilities of a simple random…

In this work we consider loop-erased random walk (LERW) and its scaling limit in three dimensions, and prove that 3D LERW parametrized by renormalized length converges to its scaling limit parametrized by some suitable measure with respect…

The determination of the Hausdorff dimension of the scaling limit of loop-erased random walk is closely related to the study of the one-point function of loop-erased random walk, i.e., the probability a loop-erased random walk passes…

Loop-erased random walk, abbreviated LERW, is one of the most well-studied critical lattice models. It is the self-avoiding random walk one gets after erasing the loops from a simple random walk in order or alternatively by considering the…

We consider loop-erased random walk (LERW) running between two boundary points of a square grid approximation of a planar simply connected domain. The LERW Green's function is the probability that the LERW passes through a given edge in the…

We measure the fractal dimension of loop-erased random walk (LERW) in 3 dimensions, and estimate that it is 1.62400 +- 0.00005. LERW is closely related to the uniform spanning tree and the abelian sandpile model. We simulated LERW on both…

We study the scaling properties of long-range loop-erased random walks (LR-LERW), where the underlying random walker performs L\'evy-flight-like jumps with a power-law step-length distribution $P(\mathbf{r})\sim |\mathbf{r}|^{-(d+\sigma)}$.…

The loop-erased random walk (LERW) in $\mathbb{Z}^4$ is the process obtained by erasing loops chronologically for simple random walk. We prove that the escape probability of the LERW renormalized by $(\log n)^{\frac{1}{3}}$ converges almost…

Loop-erased random walk and it's scaling limit, Schramm--Loewner evolution, have found numerous applications in mathematics and physics. We present a 2 dimensional analogue of LERW, the loop erased random surface. We do this by defining a 2…

We consider the occupation measure of the cut points of a simple random walk on a $d$-dimensional cubic lattice for $d = 2, 3$, and we show that the scaling limit of the occupation measure in weak topology is the natural fractal measure on…

We study the scaling limit of planar loop erased random walk (LERW) on the percolation cluster, with occupation probability $p\geq p_c$. We numerically demonstrate that the scaling limit of planar LERW$_p$ curves, for all $p>p_c$, can be…

We show that the scaling limit exists and is invariant to dilations and rotations. We give some tools that might be useful to show universality.

We show that the law of the three-dimensional uniform spanning tree (UST) is tight under rescaling in a space whose elements are measured, rooted real trees, continuously embedded into Euclidean space. We also establish that the relevant…

The infinite two-sided loop-erased random walk (LERW) is a measure on infinite self-avoiding walks that can be viewed as giving the law of the `middle part' of an infinite LERW loop going through 0 and infinity. In this note we derive…

We consider a model of loop-erased random walks on the finite pre-Sierpinski gasket which permits rigorous analysis. We prove the existence of the scaling limit and show that the path of the limiting process is almost surely self-avoiding,…

We revisit the convergence of loop-erased random walk, LERW, to SLE(2) when the curves are parametrized by capacity. We construct a coupling of the chordal version of LERW and chordal SLE(2) based on the Green's function for LERW as…

The uniform spanning tree (UST) and the loop-erased random walk (LERW) are related probabilistic processes. We consider the limits of these models on a fine grid in the plane, as the mesh goes to zero. Although the existence of scaling…