Self-intersection local times of random walks: Exponential moments in subcritical dimensions

Fix $p>1$, not necessarily integer, with $p(d-2)<d$. We study the $p$-fold self-intersection local time of a simple random walk on the lattice $Z^d$ up to time $t$. This is the $p$-norm of the vector of the walkers local times, $ell_t$. We derive precise logarithmic asymptotics of the expectation of $exp{theta_t |ell_t|_p}$ for scales $theta_t>0$ that are bounded from above, possibly tending to zero. The speed is identified in terms of mixed powers of $t$ and $theta_t$, and the precise rate is characterized in terms of a variational formula, which is in close connection to the {it Gagliardo-Nirenberg inequality}. As a corollary, we obtain a large-deviation principle for $|ell_t|_p/(t r_t)$ for deviation functions $r_t$ satisfying $t r_tggE[|ell_t|_p]$. Informally, it turns out that the random walk homogeneously squeezes in a $t$-dependent box with diameter of order $ll t^{1/d}$ to produce the required amount of self-intersections. Our main tool is an upper bound for the joint density of the local times of the walk.

Moments and distribution of the local times of a transient random walk on $Z^d$

Consider an arbitrary transient random walk on $Z^d$ with $dinN$. Pick $alphain[0,infty)$ and let $L_n(alpha)$ be the spatial sum of the $alpha$-th power of the $n$-step local times of the walk. Hence, $L_n(0)$ is the range, $L_n(1)=n+1$, and for integers $alpha$, $L_n(alpha)$ is the number of the $alpha$-fold self-intersections of the walk. We prove a strong law of large numbers for $L_n(alpha)$ as $ntoinfty$. Furthermore, we identify the asymptotic law of the local time in a random site uniformly distributed over the range. These results complement and contrast analogous results for recurrent walks in two dimensions recently derived by v{C}erny cite{Ce07}. Although these assertions are certainly known to experts, we could find no proof in the literature in this generality.