no. 1
Long-range self-avoiding walk converges to α-stable processes
Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 1, pp. 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 : 10.1214/09-AIHP350
Classification : 82B41
Keywords: self-avoiding walk, Lace expansion, α-stable processes, mean-field behavior 
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Heydenreich, Markus. Long-range self-avoiding walk converges to $\alpha $-stable processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 1, pp. 20-42. doi: 10.1214/09-AIHP350

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