Heat kernel for random walk trace on ℤ3 and ℤ4

Daisuke Shiraishi

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 4, page 1001-1024
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Shiraishi, Daisuke. "Heat kernel for random walk trace on ℤ3 and ℤ4." Annales de l'I.H.P. Probabilités et statistiques 46.4 (2010): 1001-1024. <http://eudml.org/doc/240186>.

@article{Shiraishi2010,
abstract = {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.},
author = {Shiraishi, Daisuke},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random walk in random environment; random walk trace; heat kernel estimates; cut time; loop erased random walk},
language = {eng},
number = {4},
pages = {1001-1024},
publisher = {Gauthier-Villars},
title = {Heat kernel for random walk trace on ℤ3 and ℤ4},
url = {http://eudml.org/doc/240186},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Shiraishi, Daisuke
TI - Heat kernel for random walk trace on ℤ3 and ℤ4
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 4
SP - 1001
EP - 1024
AB - 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.
LA - eng
KW - random walk in random environment; random walk trace; heat kernel estimates; cut time; loop erased random walk
UR - http://eudml.org/doc/240186
ER -

References

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  9. [9] G. F. Lawler. Intersections of Random Walks. Birkhauser, Boston, 1991. Zbl0925.60078MR1117680
  10. [10] G. F. Lawler. The logarithmic correction for loop-erased walk in four dimensions. In Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993). J. Fourier Anal. Appl. Special Issue (1995) 347–361. Zbl0889.60075MR1364896
  11. [11] G. F. Lawler. Cut times for simple random walk. Electron. J. Probab. 1 (1996) 1–24. Zbl0888.60059MR1423466
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