Heat kernel for random walk trace on 3 and 4
Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 4, p. 1001-1024

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≤kn|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.

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≤kn|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.

DOI : https://doi.org/10.1214/09-AIHP337
Classification:  82C41
Keywords: 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},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {4},
     year = {2010},
     pages = {1001-1024},
     doi = {10.1214/09-AIHP337},
     zbl = {1208.82048},
     mrnumber = {2744883},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 46 (2010) no. 4, pp. 1001-1024. doi : 10.1214/09-AIHP337. http://www.numdam.org/item/AIHPB_2010__46_4_1001_0/

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