## Brazilian Journal of Probability and Statistics

- Braz. J. Probab. Stat.
- Volume 26, Number 1 (2012), 1-55.

### An expansion for self-interacting random walks

Remco van der Hofstad and Mark Holmes

#### Abstract

We derive a perturbation expansion for general self-interacting random walks, where steps are made on the basis of the history of the path. Examples of models where this expansion applies are reinforced random walk, excited random walk, the true (weakly) self-avoiding walk, loop-erased random walk, and annealed random walk in random environment. In this paper we show that the expansion gives rise to useful formulae for the speed and variance of the random walk, when these quantities are known to exist. The results and formulae of this paper have been used elsewhere by the authors to prove monotonicity properties for the speed (in high dimensions) of excited random walk and related models, and certain models of random walk in random environment. We also derive a law of large numbers and central limit theorem (with explicit error terms) directly from this expansion, under strong assumptions on the expansion coefficients. The assumptions are shown to be satisfied by excited random walk in high dimensions with small excitation parameter, a model of reinforced random walk with underlying drift and small reinforcement parameter, and certain models of random walk in random environment under strong ellipticity conditions.

#### Article information

**Source**

Braz. J. Probab. Stat., Volume 26, Number 1 (2012), 1-55.

**Dates**

First available in Project Euclid: 11 November 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.bjps/1321043150

**Digital Object Identifier**

doi:10.1214/10-BJPS121

**Mathematical Reviews number (MathSciNet)**

MR2871279

**Zentralblatt MATH identifier**

1238.60116

**Keywords**

Self-interacting random walks excited random walk once-edge-reinforced random walk random walk in random environment lace expansion law of large numbers central limit theorem

#### Citation

van der Hofstad, Remco; Holmes, Mark. An expansion for self-interacting random walks. Braz. J. Probab. Stat. 26 (2012), no. 1, 1--55. doi:10.1214/10-BJPS121. https://projecteuclid.org/euclid.bjps/1321043150