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 $\exp \{{\theta }_t\Vert {\ell }_t{\Vert }_p\}$ 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 $\Vert {\ell }_t{\Vert }_p/(t{r}_t)$ 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

References in corpus (5)

• The universality classes in the parabolic Anderson model
• Self-intersection local time: Critical exponent, large deviations, and laws of the iterated logarithm
• Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks
• Joint density for the local times of continuous-time Markov chains
• Large deviations for intersection local times in critical dimension

Cited by in corpus (2)

• Eigenvalue fluctuations for lattice Anderson Hamiltonians
• Eigenvalue vs perimeter in a shape theorem for self-interacting random walks