Illinois Journal of Mathematics

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

Gregory F. Lawler

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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

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

First available in Project Euclid: 12 November 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]


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.

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