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

Abstract

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 ${\mathbb{Z}}^d$ up to time $t$. This is the $p$-norm of the vector of the walker's local times, ${\ell }_t$. We derive precise logarithmic asymptotics of the expectation of $\exp \{\hat{I}{¸}_t\Vert {\ell }_t{\Vert }_p\}$ for scales $\hat{I}{¸}_t>0$ that are bounded from above, possibly tending to zero. The speed is identified in terms of mixed powers of $t$ and $\hat{I}{¸}_t$, 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 $\Vert {\ell }_t{\Vert }_p/(t{r}_t)$ for deviation functions $r_t$ satisfying $t r_t\gg\E[\|\ell_t\|_p]$. 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.

15 pages. To appear in Probability Theory and Related Fields. The final publication is available at springerlink.com