Let (Xt, t≥0) be a continuous time simple random walk on ℤd (d≥3), and let lT(x) be the time spent by (Xt, t≥0) on the site x up to time T. We prove a large deviations principle for the q-fold self-intersection local time IT=∑x∈ℤdlT(x)q in the critical case q=d/(d−2). When q is integer, we obtain similar results for the intersection local times of q independent simple random walks.
"Large deviations for intersection local times in critical dimension." Ann. Probab. 38 (2) 927 - 953, March 2010. https://doi.org/10.1214/09-AOP499