Corpus ID: 214605633

Scaling limits of the three-dimensional uniform spanning tree and associated random walk

@article{Angel2020ScalingLO,
  title={Scaling limits of the three-dimensional uniform spanning tree and associated random walk},
  author={O. Angel and D. Croydon and S. Hernandez-Torres and Daisuke Shiraishi},
  journal={arXiv: Probability},
  year={2020}
}
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… Expand
A macroscopic view of two discrete random models
This thesis investigates the large-scale behaviour emerging in two discrete models: the uniform spanning tree on Z3 and the chase-escape with death process. Uniform spanning trees We consider theExpand
Logarithmic corrections to scaling in the four-dimensional uniform spanning tree
We compute the precise logarithmic corrections to mean-field scaling for various quantities describing the uniform spanning tree of the four-dimensional hypercubic lattice $\mathbb{Z}^4$. We areExpand
Uniform spanning forest on the integer lattice with drift in one coordinate.
In this article we investigate the Uniform Spanning Forest ($\mathsf{USF}$) in the nearest-neighbour integer lattice $\mathbf{Z}^{d+1} = \mathbf{Z}\times \mathbf{Z}^d$ with an assignment ofExpand
The number of spanning clusters of the uniform spanning tree in three dimensions
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 ${\mathcalExpand
PR ] 5 J ul 2 02 1 PERCOLATION TRANSITION FOR RANDOM FORESTS IN d > 3
The arboreal gas is the probability measure on (unrooted spanning) forests of a graph in which each forest is weighted by a factor β > 0 per edge. It arises as the q → 0 limit with p = βq of theExpand
Logarithmic correction to resistance.
We study the trace of the incipient infinite oriented branching random walk in $\mathbb{Z}^d \times \mathbb{Z}_+$ when the dimension is $d = 6$. Under suitable moment assumptions, we show that theExpand
Loop-Erased Random Walk as a Spin System Observable
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., theExpand

References

SHOWING 1-10 OF 53 REFERENCES
Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree
The first main result of this paper is that the law of the (rescaled) two-dimensional uniform spanning tree is tight in a space whose elements are measured, rooted real trees continuously embeddedExpand
Scaling limits of loop-erased random walks and uniform spanning trees
AbstractThe uniform spanning tree (UST) and the loop-erased random walk (LERW) are strongly related probabilistic processes. We consider the limits of these models on a fine grid in the plane, as theExpand
Scaling limits for simple random walks on random ordered graph trees
  • D. Croydon
  • Mathematics
  • Advances in Applied Probability
  • 2010
Consider a family of random ordered graph trees (T n ) n≥1, where T n has n vertices. It has previously been established that if the associated search-depth processes converge to the normalisedExpand
Conformal invariance of planar loop-erased random walks and uniform spanning trees
This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain \(D\mathop \subset \limits_ \ne \mathbb{C} \) is equal to the radial SLE2 path. In particular, theExpand
Rayleigh processes, real trees, and root growth with re-grafting
The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree.Expand
Spectral Dimension and Random Walks on the Two Dimensional Uniform Spanning Tree
We study the simple random walk on the uniform spanning tree on $${\mathbb {Z}^2}$$ . We obtain estimates for the transition probabilities of the random walk, the distance of the walk from itsExpand
Convergence of three-dimensional loop-erased random walk in the natural parametrization.
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 limitExpand
Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees)
We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measureExpand
On Brownian motion, simple paths, and loops
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 anyExpand
Volume growth and heat kernel estimates for the continuum random tree
In this article, we prove global and local (point-wise) volume and heat kernel bounds for the continuum random tree. We demonstrate that there are almost–surely logarithmic global fluctuations andExpand
...
1
2
3
4
5
...