## The Annals of Probability

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

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

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60J15 60J55: Local time and additive functionals 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

**Keywords**

Self-repelling random walk local time limit theorems anomalous diffusion

#### Citation

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