Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Long-range self-avoiding walk converges to α-stable processes

Markus Heydenreich

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Abstract

We consider a long-range version of self-avoiding walk in dimension d > 2(α ∧ 2), where d denotes dimension and α the power-law decay exponent of the coupling function. Under appropriate scaling we prove convergence to Brownian motion for α ≥ 2, and to α-stable Lévy motion for α < 2. This complements results by Slade [J. Phys. A 21 (1988) L417–L420], who proves convergence to Brownian motion for nearest-neighbor self-avoiding walk in high dimension.

Résumé

Nous considérons un modèle à longue portée de la marche aléatoire auto-évitante en dimension d > 2(α ∧ 2), où d est la dimension et α l’exposant de décroissance polynomiale de la fonction de couplage. Après un rééchelonnage approprié, nous démontrons la convergence vers un mouvement brownien pour α ≥ 2 et vers un processus de Lévy α-stable pour α < 2. Ce résultat complète celui de Slade [J. Phys. A 21 (1988) L417–L420] qui démontre la convergence vers le mouvement brownien pour une marche auto-évitante à plus proche voisin en grande dimension.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 1 (2011), 20-42.

Dates
First available in Project Euclid: 4 January 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1294170228

Digital Object Identifier
doi:10.1214/09-AIHP350

Mathematical Reviews number (MathSciNet)
MR2779395

Zentralblatt MATH identifier
1210.82055

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

Keywords
Self-avoiding walk Lace expansion α-stable processes Mean-field behavior

Citation

Heydenreich, Markus. Long-range self-avoiding walk converges to α -stable processes. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 1, 20--42. doi:10.1214/09-AIHP350. https://projecteuclid.org/euclid.aihp/1294170228


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