## Institute of Mathematical Statistics Lecture Notes - Monograph Series

- Lecture Notes--Monograph Series
- Volume 48, 2006, 66-77

### Linearly edge-reinforced random walks

Franz Merkl and Silke W. W. Rolles

#### Abstract

We review results on linearly edge-reinforced random walks. On finite graphs, the process has the same distribution as a mixture of reversible Markov chains. This has applications in Bayesian statistics and it has been used in studying the random walk on infinite graphs. On trees, one has a representation as a random walk in an independent random environment. We review recent results for the random walk on ladders: recurrence, a representation as a random walk in a random environment, and estimates for the position of the random walker.

#### Chapter information

**Source***Dynamics & Stochastics: Festschrift in honor of M. S. Keane* (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2006)

**Dates**

First available in Project Euclid: 28 November 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.lnms/1196285809

**Digital Object Identifier**

doi:10.1214/074921706000000103

**Mathematical Reviews number (MathSciNet)**

MR2306189

**Zentralblatt MATH identifier**

1125.82014

**Subjects**

Primary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments

**Keywords**

reinforced random walk random environment recurrence Bayesian statistics

**Rights**

Copyright © 2006, Institute of Mathematical Statistics

#### Citation

Merkl, Franz; Rolles, Silke W. W. Linearly edge-reinforced random walks. Dynamics & Stochastics, 66--77, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2006. doi:10.1214/074921706000000103. https://projecteuclid.org/euclid.lnms/1196285809