Scaling limits of loop-erased random walks and uniform spanning trees

@article{Schramm1999ScalingLO,
  title={Scaling limits of loop-erased random walks and uniform spanning trees},
  author={Oded Schramm},
  journal={Israel Journal of Mathematics},
  year={1999},
  volume={118},
  pages={221-288}
}
  • O. Schramm
  • Published 5 April 1999
  • Mathematics
  • Israel Journal of Mathematics
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 the mesh goes to zero. Although the existence of scaling limits is still unproven, subsequential scaling limits can be defined in various ways, and do exist. We establish some basic a.s. properties of these subsequential scaling limits in the plane. It is proved that any LERW subsequential scaling… 
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, the
Near-critical spanning forests and renormalization
We study random two-dimensional spanning forests in the plane that can be viewed both in the discrete case and in their appropriately taken scaling limits as a uniformly chosen spanning tree with
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 embedded
SCALING LIMIT OF LOOP-ERASED RANDOM WALK
The loop-erased random walk (LERW) was first studied in 1980 by Lawler as an attempt to analyze self-avoiding walk (SAW) which provides a model for the growth of a linear polymer in a good solvent.
I – Loop Erased Walks and Uniform Spanning Trees
The uniform spanning tree has had a fruitful history in probability theory. Most notably, it was the study of the scaling limit of the UST that led Oded Schramm [Sch00] to introduce the SLE process,
Loop-erased random walk and Poisson kernel on planar graphs
Lawler, Schramm and Werner showed that the scaling limit of the loop-erased random walk on $\mathbb{Z}^2$ is $\mathrm{SLE}_2$. We consider scaling limits of the loop-erasure of random walks on other
Scaling limits for minimal and random spanning trees in two dimensions
A general formulation is presented for continuum scaling limits of stochastic spanning trees. A spanning tree is expressed in this limit through a consistent collection of subtrees, which includes a
Scaling limits of critical systems in random geometry
This thesis focusses on the properties of, and relationships between, several fundamental objects arising from critical physical models. In particular, we consider Schramm–Loewner evolutions, the
The dimension of the SLE curves
Let γ be the curve generating a Schramm–Loewner Evolution (SLE) process, with parameter κ ≥ 0. We prove that, with probability one, the Haus-dorff dimension of γ is equal to Min(2, 1 + κ/8).
...
...

References

SHOWING 1-10 OF 73 REFERENCES
Scaling limits for minimal and random spanning trees in two dimensions
A general formulation is presented for continuum scaling limits of stochastic spanning trees. A spanning tree is expressed in this limit through a consistent collection of subtrees, which includes a
Continuum Limits for Critical Percolation and Other Stochastic Geometric Models
The talk presented at ICMP 97 focused on the scaling limits of critical percolation models, and some other systems whose salient features can be described by collections of random lines. In the
Exact partition functions and correlation functions of multiple Hamiltonian walks on the Manhattan lattice
This is a general and exact study of multiple Hamiltonian walks (HAW) filling the two-dimensional (2D) Manhattan lattice. We generalize the original exact solution for a single HAW by Kasteleyn to a
Percolation ?
572 NOTICES OF THE AMS VOLUME 53, NUMBER 5 Percolation is a simple probabilistic model which exhibits a phase transition (as we explain below). The simplest version takes place on Z2, which we view
Treelike structures arising from continua and convergence groups
In this paper, we develop the theory of a very general class of treelike structures based on a simple set of betweenness axioms. Within this framework, we explore connections between more familiar
Loop-Erased Random Walk
Loop-erased random walk (LERW) is a process obtained from erasing loops from simple random walk. This paper reviews some of the results and conjectures about LERW. In particular, we discuss the
Exact determination of the percolation hull exponent in two dimensions.
TLDR
It is argued finally that the different fractal dimensions observed recently by Grossman and Aharony, who modified the definition of the hull, are all equal to ${D}_{e}=\frac{4}{3}$.
Conformal invariance in two-dimensional percolation
The word percolation, borrowed from the Latin, refers to the seeping or oozing of a liquid through a porous medium, usually to be strained. In this and related senses it has been in use since the
...
...