Large deviations for intersection local times in critical dimension

Abstract

Let (X_t, t≥0) be a continuous time simple random walk on ℤ^d (d≥3), and let l_T(x) be the time spent by (X_t, t≥0) on the site x up to time T. We prove a large deviations principle for the q-fold self-intersection local time I_T=∑_x∈ℤ^dl_T(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.