Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Scaling limits of loop-erased random walks and uniform spanning trees   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.

30
The categories of this paper are: q-alg
16
The categories of this paper are: stat
36
The categories of this paper are: cmp-lg
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The categories of this paper are: chem-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: comp-gas
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The categories of this paper are: supr-con
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The categories of this paper are: adap-org
6
The categories of this paper are: astro-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: eess
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The categories of this paper are: econ
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-th
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The categories of this paper are: alg-geom
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The categories of this paper are: funct-an