A Lower Bound on the Growth Exponent for Loop-Erased Random Walk in Two Dimensions

Free access article

Issue		ESAIM: PS Volume 3, 1999


Page(s)		1 - 21
DOI		10.1051/ps:1999100

DOI: 10.1051/ps:1999100

ESAIM: PS, April 1999, Vol. 3, p. 1-21

A Lower Bound on the Growth Exponent for Loop-Erased Random Walk in Two Dimensions

Minoration de l'exposant de croissance pour une marche aléatoire à boucles effacées en dimension deux

Gregory F. Lawler
Duke University Durham, NC 27708-0320, USA; (jose@math.duke.edu)

Received April 10, 1998. Revised September 21, 1998.

Abstract: The growth exponent $\alpha$ for loop-erased or Laplacian random walk on the integer lattice is defined by saying that the expected time to reach the sphere of radius n is of order $n^\alpha$ . We prove that in two dimensions, the growth exponent is strictly greater than one. The proof uses a known estimate on the third moment of the escape probability and an improvement on the discrete Beurling projection theorem.

Résumé: L'exposant de croissance $\alpha$ pour la marche aléatoire à boucles effacées ou "laplacienne" sur le réseau Z^d est défini de la manière suivante : le nombre moyen de pas au moment où la marche issue de l'origine atteint la sphère de rayon n est d'ordre $n^\alpha$ lorsque n tend vers l'infini. Nous montrons que lorsque d=2, l'exposant de croissance est strictement supérieur à 1. La preuve utilise une estimation connue concernant le moment d'ordre trois de la probabilité de fuite, ainsi qu'un raffinement de la version discrétisée du théorème de projection de Beurling.

Keywords and phrases: loop-erased walk, Beurling projection theorem

AMS Subject Classification: 60J15

Article without figures.

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