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.
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}, pages = {20--42}, publisher = {Gauthier-Villars}, volume = {47}, number = {1}, year = {2011}, doi = {10.1214/09-AIHP350}, zbl = {1210.82055}, language = {en}, url = {http://www.numdam.org/articles/10.1214/09-AIHP350/} }
TY - JOUR AU - Heydenreich, Markus TI - Long-range self-avoiding walk converges to $\alpha $-stable processes JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2011 SP - 20 EP - 42 VL - 47 IS - 1 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/09-AIHP350/ DO - 10.1214/09-AIHP350 LA - en ID - AIHPB_2011__47_1_20_0 ER -
%0 Journal Article %A Heydenreich, Markus %T Long-range self-avoiding walk converges to $\alpha $-stable processes %J Annales de l'I.H.P. Probabilités et statistiques %D 2011 %P 20-42 %V 47 %N 1 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/09-AIHP350/ %R 10.1214/09-AIHP350 %G en %F 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/articles/10.1214/09-AIHP350/
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