In high dimensions two independent simple random walks have only a finite number of intersections. I describe the main result obtained in a joint paper with Xia Chen in which we determine the exact upper tail behaviour of the intersection local time.
@article{ACIRM_2010__2_1_27_0,
author = {Peter M\"orters},
title = {Upper tail asymptotics for the intersection local times of random walks in high dimensions},
journal = {Actes des rencontres du CIRM},
publisher = {CIRM},
volume = {2},
number = {1},
year = {2010},
pages = {27-29},
doi = {10.5802/acirm.20},
language = {en},
url = {https://acirm.centre-mersenne.org/item/ACIRM_2010__2_1_27_0}
}
Mörters, Peter. Upper tail asymptotics for the intersection local times of random walks in high dimensions. Actes des rencontres du CIRM, Volume 2 (2010) no. 1, pp. 27-29. doi : 10.5802/acirm.20. https://acirm.centre-mersenne.org/item/ACIRM_2010__2_1_27_0/
[1] X. Chen and P. Mörters. Upper tails for intersection local times of random walks in supercritial dimensions. Journal of the London Mathematical Society, 79 (2009) 186-210.
[2] A. Dvoretzky and P. Erdős. Some problems on random walk in space. In: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950. University of California Press, Berkeley and Los Angeles, pp. 353–367, 1951.
[3] K.M. Khanin, A.E. Mazel, S.B. Shlosman and Ya.G. Sinai. Loop condensation effects in the behavior of random walks. In: Markov processes and applications, Birkhäuser, pp. 167–184, 1994.
[4] W. König and P. Mörters. Brownian intersection local times: Upper tail asymptotics and thick points. Annals of Probability, 30 (2002) 1605-1656.