## The Annals of Probability

### The "True" Self-Avoiding Walk with Bond Repulsion on $\mathbb{Z}$: Limit Theorems

Balint Toth

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 23, Number 4 (1995), 1523-1556.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176987793

Digital Object Identifier
doi:10.1214/aop/1176987793

Mathematical Reviews number (MathSciNet)
MR1379158

Zentralblatt MATH identifier
0852.60083

JSTOR
Toth, Balint. The "True" Self-Avoiding Walk with Bond Repulsion on $\mathbb{Z}$: Limit Theorems. Ann. Probab. 23 (1995), no. 4, 1523--1556. doi:10.1214/aop/1176987793. https://projecteuclid.org/euclid.aop/1176987793