Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Hausdorff dimension of the scaling limit of loop-erased random walk in three dimensions

Daisuke Shiraishi

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Let $M_{n}$ be the length (number of steps) of the loop-erasure of a simple random walk up to the first exit from a ball of radius $n$ centered at its starting point. It is shown in (Ann. Probab. 46 (2) (2018) 687–774) that there exists $\beta\in(1,\frac{5}{3}]$ such that $E(M_{n})$ is of order $n^{\beta}$ in 3 dimensions. In the present article, we show that the Hausdorff dimension of the scaling limit of the loop-erased random walk in 3 dimensions is equal to $\beta$ almost surely.


Soit $M_{n}$ la longueur (nombre de pas) d’une marche aléatoire simple à boucles effacées considérée jusqu’à la première sortie d’une boule de rayon $n$ centrée en son point de départ. Il est démontré dans (Ann. Probab. 46 (2) (2018) 687–774) qu’il existe $\beta\in(1,\frac{5}{3}]$ tel que $E(M_{n})$ est d’ordre $n^{\beta}$ en dimension 3. Dans le présent article, nous montrons que la dimension de Hausdorff de la limite d’échelle de la marche aléatoire effacée en dimension $3$ est égale à $\beta$ presque sûrement.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 2 (2019), 791-834.

Received: 28 April 2016
Revised: 22 January 2018
Accepted: 17 March 2018
First available in Project Euclid: 14 May 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Loop-erased random walk Scaling limit Hausdorff dimension


Shiraishi, Daisuke. Hausdorff dimension of the scaling limit of loop-erased random walk in three dimensions. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 2, 791--834. doi:10.1214/18-AIHP899.

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