Open AccessPosted Content
Scaling limits of loop-erased random walks and uniform spanning trees
Reads0
Chats0
TLDR
In this paper, it was shown that the scaling limit of the LERW process is conformally invariant in 2-dimensional space, and that the UST scaling limit is a topological tree.Abstract:
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 limits is still unproven, subsequential scaling limits can be defined in various ways, and do exist. We establish some basic a.s. properties of the subsequential scaling limits in the plane. It is proved that any LERW subsequential scaling limit is a simple path, and that the trunk of any UST subsequential scaling limit is a topological tree, which is dense in the plane.
The scaling limits of these processes are conjectured to be conformally invariant in 2 dimensions. We make a precise statement of the conformal invariance conjecture for the LERW, and show that this conjecture implies an explicit construction of the scaling limit, as follows. Consider the Loewner differential equation
${\partial f\over\partial t}
= z {\zeta(t)+z \over \zeta(t)-z} {\partial f\over\partial z}$
with boundary values $f(z,0)=z$, in the range $z\in\U=\{w\in\C\st |w|<1\}$, $t\le 0$. We choose $\zeta(t):= \B(-2t)$, where $\B(t)$ is Brownian motion on $\partial \U$ starting at a random-uniform point in $\partial \U$. Assuming the conformal invariance of the LERW scaling limit in the plane, we prove that the scaling limit of LERW from 0 to $\partial\U$ has the same law as that of the path $f(\zeta(t),t)$. We believe that a variation of this process gives the scaling limit of the boundary of macroscopic critical percolation clusters.read more
Citations
More filters
Posted Content
Continuous Move From BTW to Manna Model
TL;DR: In this article, the authors consider the BTW model perturbed by random-direction anisotropy with strength factor \epsilon ranging from 0 to 1 corresponding to BTW and Manna model.
Posted Content
Water Propagation in the Porous Media, Self-Organized Criticality and Ising Model
Morteza N. Najafi,Mojtaba Ghaedi +1 more
TL;DR: In this article, the authors proposed the Ising model to study the propagation of water in 2D petroleum reservoir in which each bond between its pores has the probability $p$ of being activated.
Dissertation
Exposants géométriques des modèles de boucles dilués et idempotents des TL-modules de la chaîne de spins XXZ
Journal ArticleDOI
Probabilistic construction of Toda Conformal Field Theories
TL;DR: In this article , the authors use the path integral formulation of these models to provide a rigorous mathematical construction of Toda conformal field theories based on probability theory and recover expected properties of the theory such as the Weyl anomaly formula with respect to the change of background metric by a conformal factor.
Book ChapterDOI
Conformal Spiral Multifractals
TL;DR: In this article, the exact joint multifractal distribution for the scaling and spiraling of electrostatic potential lines near any conformally invariant scaling curve is derived in two dimensions.
References
More filters
Proceedings ArticleDOI
Generating random spanning trees more quickly than the cover time
TL;DR: This paper gives a new algorithm for generating random spanning trees of an undirected graph that is easy to code up, has small running time constants, and has a nice proof that it generates trees with the right probabilities.
Journal ArticleDOI
Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I.
Journal ArticleDOI
Critical percolation in finite geometries
TL;DR: In this paper, the methods of conformal field theory are used to compute the crossing probabilities between segments of the boundary of a compact two-dimensional region at the percolation threshold.
Journal ArticleDOI
Exact Determination of the Percolation Hull Exponent in Two Dimensions
TL;DR: 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 ${D}_{e}=\frac{4}{3}$.