Nous étudions la marche aléatoire simple sur l'ensemble des points visités par une marche aléatoire simple sur ℤ3 et ℤ4. En dimension quatre, nous établissons des bornes presque sûres pour le noyau de la chaleur de X et pour max0≤k≤n|Xk| qui nécessitent des termes correctifs logarithmiques. En dimension trois, nous montrons que X à un comportement non diffusif presque sûrement. Pour démontrer ces résultats, nous obtenons des estimées asymptotiques pour les temps de coupure de la marche aléatoire simple et pour la marche à boucles effacées qui sont intéressantes en elles-mêmes.
We study the simple random walk X on the range of simple random walk on ℤ3 and ℤ4. In dimension four, we establish quenched bounds for the heat kernel of X and max0≤k≤n|Xk| which require extra logarithmic correction terms to the higher-dimensional case. In dimension three, we demonstrate anomalous behavior of X at the quenched level. In order to establish these estimates, we obtain several asymptotic estimates for cut times of simple random walk and asymptotic estimates for loop-erased random walk, which are of independent interest.
Mots clés : random walk in random environment, random walk trace, heat kernel estimates, cut time, loop erased random walk
@article{AIHPB_2010__46_4_1001_0, author = {Shiraishi, Daisuke}, title = {Heat kernel for random walk trace on $\mathbb {Z}^3$ and $\mathbb {Z}^4$}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {1001--1024}, publisher = {Gauthier-Villars}, volume = {46}, number = {4}, year = {2010}, doi = {10.1214/09-AIHP337}, mrnumber = {2744883}, zbl = {1208.82048}, language = {en}, url = {http://www.numdam.org/articles/10.1214/09-AIHP337/} }
TY - JOUR AU - Shiraishi, Daisuke TI - Heat kernel for random walk trace on $\mathbb {Z}^3$ and $\mathbb {Z}^4$ JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2010 SP - 1001 EP - 1024 VL - 46 IS - 4 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/09-AIHP337/ DO - 10.1214/09-AIHP337 LA - en ID - AIHPB_2010__46_4_1001_0 ER -
%0 Journal Article %A Shiraishi, Daisuke %T Heat kernel for random walk trace on $\mathbb {Z}^3$ and $\mathbb {Z}^4$ %J Annales de l'I.H.P. Probabilités et statistiques %D 2010 %P 1001-1024 %V 46 %N 4 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/09-AIHP337/ %R 10.1214/09-AIHP337 %G en %F AIHPB_2010__46_4_1001_0
Shiraishi, Daisuke. Heat kernel for random walk trace on $\mathbb {Z}^3$ and $\mathbb {Z}^4$. Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 4, pp. 1001-1024. doi : 10.1214/09-AIHP337. http://www.numdam.org/articles/10.1214/09-AIHP337/
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