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October, 1995 The "True" Self-Avoiding Walk with Bond Repulsion on $\mathbb{Z}$: Limit Theorems
Balint Toth
Ann. Probab. 23(4): 1523-1556 (October, 1995). DOI: 10.1214/aop/1176987793


The "true" self-avoiding walk with bond repulsion is a nearest neighbor random walk on $\mathbb{Z}$, for which the probability of jumping along a bond of the lattice is proportional to $\exp(-g \cdot$ number of previous jumps along that bond). First we prove a limit theorem for the distribution of the local time process of this walk. Using this result, later we prove a local limit theorem, as $A \rightarrow \infty$, for the distribution of $A^{-2/3}X_{\theta_{s/A}}$, where $\theta_{s/A}$ is a random time distributed geometrically with mean $e^{-s/A}(1 - e^{-s/A})^{-1} = A/s + O(1)$. As a by-product we also obtain an apparently new identity related to Brownian excursions and Bessel bridges.


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Balint Toth. "The "True" Self-Avoiding Walk with Bond Repulsion on $\mathbb{Z}$: Limit Theorems." Ann. Probab. 23 (4) 1523 - 1556, October, 1995.


Published: October, 1995
First available in Project Euclid: 19 April 2007

zbMATH: 0852.60083
MathSciNet: MR1379158
Digital Object Identifier: 10.1214/aop/1176987793

Primary: 60F05
Secondary: 60J15 , 60J55 , 82C41

Keywords: Anomalous diffusion , limit theorems , Local time , Self-repelling random walk

Rights: Copyright © 1995 Institute of Mathematical Statistics

Vol.23 • No. 4 • October, 1995
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