Large deviations for intersection local times in critical dimension

Let $(X_t,t\ge 0)$ be a continuous time simple random walk on ${\mathbb{Z}}^d$ ($d\ge 3$), and let $l_T(x)$ be the time spent by $(X_t,t\ge 0)$ on the site $x$ up to time $T$. We prove a large deviations principle for the $q$-fold self-intersection local time $I_T={\sum }_{x\in {\mathbb{Z}}^d}{l}_T(x{)}^q$ in the critical case $q=\frac{d}{d-2}$. When $q$ is integer, we obtain similar results for the intersection local times of $q$ independent simple random walks.