Let $M_n$ be the number of steps of the loop-erasure of a simple random walk
on $\mathbb{Z}^2$ from the origin to the circle of radius $n$. We relate the
moments of $M_n$ to $Es(n)$, the probability that a random walk and an
independent loop-erased random walk both started at the origin do not intersect
up to leaving the ball of radius $n$. This allows us to show that there exists
$C$ such that for all $n$ and all $k=1,2,...,\mathbf{E}[M_n^k]\leq
C^kk!\mathbf{E}[M_n]^k$ and hence to establish exponential moment bounds for
$M_n$. This implies that there exists $c>0$ such that for all $n$ and all
$\lambda\geq0$, \[\mathbf{P}\{M_n>\lambda\mathbf{E}[M_n]\}\leq2e^{-c\lambda}.\]
Using similar techniques, we then establish a second moment result for a
specific conditioned random walk which enables us to prove that for any
$\alpha<4/5$, there exist $C$ and $c'>0$ such that for all $n$ and $\lambda>0$,
\[\mathbf{P}\{M_n<\lambda^{-1}\mathbf{E}[M_n]\}\leq Ce^{-c'\lambda
^{\alpha}}.\]Comment: Published in at http://dx.doi.org/10.1214/10-AOP539 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
We give a new proof of a result of Kenyon that the growth exponent for loop-erased random walks in two dimensions is 5/4. The proof uses the convergence of LERW to Schramm-Loewner evolution with parameter 2, and is valid for irreducible bounded symmetric random walks on any discrete lattice of R 2 .
We study simple random walk on the uniform spanning tree on Z 2 . We obtain estimates for the transition probabilities of the random walk, the distance of the walk from its starting point after n steps, and exit times of both Euclidean balls and balls in the intrinsic graph metric. In particular, we prove that the spectral dimension of the uniform spanning tree on Z 2 is 16/13 almost surely.
We outline a strategy for showing convergence of loop-erased random walk on the Z 2 square lattice to SLE(2), in the supremum norm topology that takes the time parametrization of the curves into account. The discrete curves are parametrized so that the walker moves at a constant speed determined by the lattice spacing, and the SLE(2) curve has the recently introduced natural time parametrization. Our strategy can be seen as an extension of the one used by Lawler, Schramm, and Werner to prove convergence modulo time parametrization. The crucial extra step is showing that the expected occupation measure of the discrete curve, properly renormalized by the chosen time parametrization, converges to the occupation density of the SLE(2) curve, the so-called SLE Green's function. Although we do not prove this convergence, we rigorously establish some partial results in this direction including a new loop-erased random walk estimate. t → Y n (t) := 1 n X(σ n (t))converges weakly as n → ∞ with respect to the topology of the supremum norm on curves, and 2010 Mathematics Subject Classification. 60J67, 82B31, 82B41.
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