We prove sharp estimates on the expected number of windings of a simple random walk on the square or triangular lattice. This gives new lower bounds on the averaged Dehn function, which measures the expected area needed to fill a random curve with a disc.
Le principal résultat de cet article donne un équivalent précis de l'espérance du nombre total de tours effectués par la marche aléatoire simple sur ℤ2 ou sur le réseau triangulaire. Comme corollaire, nous obtenons une nouvelle borne inférieure de la fonction de Dehn moyennée sur ℤd, d ≥ 2, qui mesure l'aire moyenne du disque remplissant de manière optimale une courbe de longueur donnée.
Keywords: simple random walk, winding number, averaged Dehn function
@article{AIHPB_2011__47_1_130_0, author = {Schapira, Bruno and Young, Robert}, title = {Windings of planar random walks and averaged {Dehn} function}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {130--147}, publisher = {Gauthier-Villars}, volume = {47}, number = {1}, year = {2011}, doi = {10.1214/10-AIHP365}, mrnumber = {2779400}, language = {en}, url = {http://www.numdam.org/articles/10.1214/10-AIHP365/} }
TY - JOUR AU - Schapira, Bruno AU - Young, Robert TI - Windings of planar random walks and averaged Dehn function JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2011 SP - 130 EP - 147 VL - 47 IS - 1 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/10-AIHP365/ DO - 10.1214/10-AIHP365 LA - en ID - AIHPB_2011__47_1_130_0 ER -
%0 Journal Article %A Schapira, Bruno %A Young, Robert %T Windings of planar random walks and averaged Dehn function %J Annales de l'I.H.P. Probabilités et statistiques %D 2011 %P 130-147 %V 47 %N 1 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/10-AIHP365/ %R 10.1214/10-AIHP365 %G en %F AIHPB_2011__47_1_130_0
Schapira, Bruno; Young, Robert. Windings of planar random walks and averaged Dehn function. Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 1, pp. 130-147. doi : 10.1214/10-AIHP365. http://www.numdam.org/articles/10.1214/10-AIHP365/
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