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Scaling limits of loop-erased random walks and uniform spanning trees
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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
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Book
Probability on Trees and Networks
Russell Lyons,Yuval Peres +1 more
TL;DR: In this article, the authors present a state-of-the-art account of probability on networks, including percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks.
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Conformally Invariant Processes in the Plane
TL;DR: The Schramm-Loewner evolution (SLE) is a Loewner chain with a Brownian motion input as discussed by the authors, which is a variant of the SLE.
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Conformal invariance of planar loop-erased random walks and uniform spanning trees
TL;DR: In this article, it was shown that the scaling limit of a loop-erased random walk in a simply connected domain is equal to the radial SLE2 path, and that the limit exists and is conformally invariant.
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A survey of random processes with reinforcement
TL;DR: The models surveyed in this paper include generalized Polya urns, reinforced random walks, interacting urn models, and continuous reinforced processes, with a focus on methods and results, with sketches provided of some proofs.
References
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Self-Avoiding Walks
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