Large deviations estimates for self-intersection local times for simple random walk in $${\mathbb{Z}}^3$$

Abstract

We obtain large deviations estimates for the self-intersection local times for a simple random walk in dimension 3. Also, we show that the main contribution to making the self-intersection large, in a time period of length n, comes from sites visited less than some power of log(n). This is opposite to the situation in dimensions larger or equal to 5. Finally, we present an application of our estimates to moderate deviations for random walk in random sceneries.