## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 22 (2017), paper no. 66, 19 pp.

### Scaling limit of the uniform prudent walk

Nicolas Pétrélis, Rongfeng Sun, and Niccolò Torri

#### Abstract

We study the 2-dimensional uniform prudent self-avoiding walk, which assigns equal probability to all nearest-neighbor self-avoiding paths of a fixed length that respect the prudent condition, namely, the path cannot take any step in the direction of a previously visited site. The uniform prudent walk has been investigated with combinatorial techniques in [3], while another variant, the kinetic prudent walk has been analyzed in detail in [2]. In this paper, we prove that the $2$-dimensional uniform prudent walk is ballistic and follows one of the $4$ diagonals with equal probability. We also establish a functional central limit theorem for the fluctuations of the path around the diagonal.

#### Article information

**Source**

Electron. J. Probab., Volume 22 (2017), paper no. 66, 19 pp.

**Dates**

Received: 21 February 2017

Accepted: 26 July 2017

First available in Project Euclid: 7 September 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1504749661

**Digital Object Identifier**

doi:10.1214/17-EJP87

**Mathematical Reviews number (MathSciNet)**

MR3698735

**Zentralblatt MATH identifier**

1377.82028

**Subjects**

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

Secondary: 60F05: Central limit and other weak theorems 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

**Keywords**

self-avoiding walk prudent walk scaling limits

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Pétrélis, Nicolas; Sun, Rongfeng; Torri, Niccolò. Scaling limit of the uniform prudent walk. Electron. J. Probab. 22 (2017), paper no. 66, 19 pp. doi:10.1214/17-EJP87. https://projecteuclid.org/euclid.ejp/1504749661