IJPAM: Volume 31, No. 1 (2006)


Wei-Shih Yang$^1$, Aklilu Zeleke$^2$
$^1$Department of Mathematics
Temple University
638 Wachman Hall, 1805 N. Broad Street, Philadelphia, PA 19122, USA
e-mail: yang@temple.edu
$^2$Lyman Briggs School of Science and
Department of Statistics and Probability
Michigan State University
East Lansing, MI 48825-1107, USA
e-mail: zeleke@msu.edu

Abstract.Let $S_{n}$ be a simple random walk (SRW) defined on $Z^{3}$. We construct a stochastic process from $S_{n}$ by erasing loops of length at most $N^{\alpha}$, where $\alpha \in (0, \infty]$ and $N$ is the scaling parameter that will be taken to infinity in determining the limiting distribution. We call this process the $N^{\alpha}$ loop erased walk ($N^{\alpha}$ LEW). Under some assumptions we will prove that for $0 < \alpha <
\frac{1}{1+2\zeta}$, the limiting distribution is Gaussian. Here $\zeta$ is the intersection exponent of random walks in $Z^{3}.$ For $\alpha > 2$ the limiting distribution is equal to the limiting distribution of the loop erased walk ( $\alpha = \infty$). It is known that $.25 < \zeta < .5 $. We conjecture that for $\alpha < 2$, the limiting distribution of $N^{\alpha}$ LEW is Gaussian and hence the critical value is $\alpha _c =2$. Our result implies that the complexity of simulating an $N$-step loop erased walk on $Z^3$ has a deterministic uniform upper bound $O(N^3)$ and lower bound $O(N^{3/2})$.

Received: July 5, 2006

AMS Subject Classification: 60G50, 82B41

Key Words and Phrases: loop erased walk, intersection of random walks, complexity of self-avoiding paths

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 31
Issue: 1