## Illinois Journal of Mathematics

- Illinois J. Math.
- Volume 50, Number 1-4 (2006), 701-746.

### The Laplacian-$b$ random walk and the Schramm-Loewner evolution

#### Abstract

The Laplacian-$b$ random walk is a measure on self-avoiding paths that at each step has translation probabilities weighted by the $b$th power of the probability that a simple random walk avoids the path up to that point. We give a heuristic argument as to what the scaling limit should be and call this process the Laplacian-$b$ motion, $LM_b$. In simply connected domains, this process is the Schramm-Loewner evolution with parameter $\kappa = 6/(2b+1)$. In non-simply connected domains, it corresponds to the harmonic random Loewner chains as introduced by Zhan.

#### Article information

**Source**

Illinois J. Math., Volume 50, Number 1-4 (2006), 701-746.

**Dates**

First available in Project Euclid: 12 November 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ijm/1258059489

**Digital Object Identifier**

doi:10.1215/ijm/1258059489

**Mathematical Reviews number (MathSciNet)**

MR2247843

**Zentralblatt MATH identifier**

1128.60069

**Subjects**

Primary: 60J65: Brownian motion [See also 58J65]

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

#### Citation

Lawler, Gregory F. The Laplacian-$b$ random walk and the Schramm-Loewner evolution. Illinois J. Math. 50 (2006), no. 1-4, 701--746. doi:10.1215/ijm/1258059489. https://projecteuclid.org/euclid.ijm/1258059489