On the tail of the branching random walk local time

Consider a critical branching random walk on ${\mathbb{Z}}^d$, $d\ge 1$, started with a single particle at the origin, and let $L(x)$ be the total number of particles that ever visit a vertex $x$. We study the tail of $L(x)$ under suitable conditions on the offspring distribution. In particular, our results hold if the offspring distribution has an exponential moment.