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In this work, we consider the scaling limit of loop-erased random walk (LERW) in three dimensions and prove that the limiting occupation measure is equivalent to its $\beta$-dimensional Minkowski content, where $\beta \in (1, 5/3]$ is its Hausdorff dimension. In doing this we also establish the existence of the two-point function and provide some sharp estimates on one-point function and ball-hitting probabilities for 3D LERW in any scale, which is a considerable improvement of previous results., Comment: 50 pages

We derive sub-Gaussian bounds for the annealed transition density of the simple random walk on a high-dimensional loop-erased random walk. The walk dimension that appears in these is the exponent governing the space-time scaling of the process with respect to the extrinsic Euclidean distance, which contrasts with the exponent given by the intrinsic graph distance that appears in the corresponding quenched bounds. We prove our main result using novel pointwise Gaussian estimates on the distribution of the high-dimensional loop-erased random walk., Comment: 36 pages, 7 figures

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 the Brownian cut points defined via its Minkowski content., Comment: 57 pages, 12 figures

We investigate the number of holes created by an ``earthworm'' moving on the two-dimensional integer lattice. The earthworm is modeled by a simple random walk. At the initial time, all vertices are filled with grains of soil except for the position of the earthworm. At each step, the earthworm pushes the soil in the direction of its motion. It leaves a hole (an empty vertex with no grain of soil) behind it. If there are holes in front of the earthworm (in the direction of its step), the closest hole is filled with a grain of soil. Thus the number of holes increases by 1 or remains unchanged at every step. We show that the number of holes is at least $\mathcal{O}(n^{3/4})$ after $n$ steps.

Let $\mathcal{U}$ be the uniform spanning tree on $\mathbb{Z}^{3}$. We show the occurrence of log-logarithmic fluctuations around the leading order for the volume of intrinsic balls in $\mathcal{U}$. As an application, we obtain similar fluctuations for the quenched heat kernel of the simple random walk on $\mathcal{U}$., Comment: 39 pages, 20 figures

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 almost surely., Comment: 37 pages, 7 figures

We establish scaling limits for the random walk whose state space is the range of a simple random walk on the four-dimensional integer lattice. These concern the asymptotic behaviour of the graph distance from the origin and the spatial location of the random walk in question. The limiting processes are the analogues of those for higher-dimensional versions of the model, but additional logarithmic terms in the scaling factors are needed to see these. The proof applies recently developed machinery relating the scaling of resistance metric spaces and stochastic processes, with key inputs being natural scaling statements for the random walk's invariant measure, the associated effective resistance metric, the graph distance, and the cut times for the underlying simple random walk.

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 laws actually converge along a particular scaling sequence. The techniques that we use to establish these results are further applied to obtain various properties of the intrinsic metric and measure of any limiting space, including showing that the Hausdorff dimension of such is given by $3/\beta$, where $\beta\approx 1.624\dots$ is the growth exponent of three-dimensional loop-erased random walk. Additionally, we study the random walk on the three-dimensional uniform spanning tree, deriving its walk dimension (with respect to both the intrinsic and Euclidean metric) and its spectral dimension, demonstrating the tightness of its annealed law under rescaling, and deducing heat kernel estimates for any diffusion that arises as a scaling limit., Comment: 78 pages, 11 figures. V2: Several minor corrections. The order of Section 3 was rearranged

Let ${\mathcal U}_{\delta}$ be the uniform spanning tree on $\delta \mathbb{Z}^{3}$. A spanning cluster of ${\mathcal U}_{\delta}$ is a connected component of the restriction of ${\mathcal U}_{\delta}$ to the unit cube $[0,1]^{3}$ that connects the left face $\{ 0 \} \times [0,1]^{2}$ to the right face $\{ 1 \} \times [0,1]^{2}$. In this note, we will prove that the number of the spanning clusters is tight as $\delta \to 0$, which resolves an open question raised by Benjamini (1999)., Comment: 8 pages, 1 figure

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 to the uniform convergence topology in the lattice size scaling limit. Our result greatly improves the work (Acta Math. 199(1):29-152) of Gady Kozma which establishes the weak convergence of the rescaled trace of 3D LERW towards a random compact set with respect to the Hausdorff distance., Comment: 90 pages, 19 figures, 5 tables. Revised version

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 dyadic scales. These estimates will be crucial to the characterization of the convergence of LERW to its scaling limit in natural parametrization. As a step in the proof, we also obtain a coupling of two pairs of LERW and SRW with different starting points conditioned to avoid each other., Comment: 39 pages, 2 figures

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 that $E (M_{n} )$ is of order $n^{\beta}$ in 3 dimensions. In the present article, we show that the Hausdorff dimension of the scaling limit of the loop-erased random walk in 3 dimensions is equal to $\beta$ almost surely., Comment: 39 pages, 1 figure

We provide a decomposition of the trace of the Brownian motion into a simple path and an independent Brownian soup of loops that intersect the simple path. More precisely, we prove that any subsequential scaling limit of the loop erased random walk is a simple path (a new result in three dimensions), which can be taken as the simple path of the decomposition. In three dimensions, we also prove that the Hausdorff dimension of any such subsequential scaling limit lies in $(1,\frac53]$. We conjecture that our decomposition characterizes uniquely the law of the simple path. If so, our results would give a new strategy to the existence of the scaling limit of the loop erased random walk and its rotational invariance., Comment: 38 pages

We show the existence of the growth exponent for loop-erased random walk in three dimensions., Comment: 59 pages, 3 figures

Let $S^{1},S^{2}$ be independent simple random walks in $\mathbb{Z}^{d}$ ($d=2,3$) started at the origin. We construct two-sided random walk paths conditioned that $S^{1}[0,\infty) \cap S^{2}[1, \infty) = \emptyset$., Comment: 25 pages