Long-range self-avoiding walk converges to α-stable processes
Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 1, p. 20-42

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.

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.

DOI : https://doi.org/10.1214/09-AIHP350
Classification:  82B41
Keywords: self-avoiding walk, Lace expansion, α-stable processes, mean-field behavior
@article{AIHPB_2011__47_1_20_0,
     author = {Heydenreich, Markus},
     title = {Long-range self-avoiding walk converges to $\alpha $-stable processes},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {1},
     year = {2011},
     pages = {20-42},
     doi = {10.1214/09-AIHP350},
     zbl = {1210.82055},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2011__47_1_20_0}
}
Heydenreich, Markus. Long-range self-avoiding walk converges to $\alpha $-stable processes. Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 1, pp. 20-42. doi : 10.1214/09-AIHP350. http://www.numdam.org/item/AIHPB_2011__47_1_20_0/

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