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.

Random walk in random scenery and self-intersection local times in dimensions d ≥ 5

“…Remark 1.7 Recently, the first author has obtained in [1] large deviations estimates for the SILT in d = 3, and has shown that the main contribution in making the SILT large, in a time period of length n, comes from sites whose local times is less than some power of log(n). Also, a diagram for the ζ -exponent for the deviations for RWRS is given in d = 3.…”

“…About R 1 We first obtain the existence of some exponential moments for J (l) . For each l ≤ l * , k = 1, .…”

“…We then divide n (1) 1 (resp. n (1) 2 ) similarly into n (2) 1 and n (2) 2 (resp. n (2) 3 and n (2) 4 ), observing that max{n (2) k , k = 1, .…”

“…The displacement along e x consists of two independent parts: a sum of i.i.d. random variables α 1 + · · · + α n , and a sum of dependent random variables η(S 0 ) + · · · + η(S n ), which is our RWRS denoted by X n in (1). Before dealing with RWRS, let us first recall the classical estimates for P(Y 1 + · · · + Y n > n β ), where β > 1/2 and the {Y n , n ∈ N} are centered i.i.d.…”

Moderate deviations for a random walk in random scenery

“…Eξ(0) = 0, with variance σ 2 > 0, and satisfies E|ξ(0)| 3 < ∞ and Cramér's condition, E e θξ(0) < ∞ for some θ > 0. (1) We are interested in the cumulative sceneries as seen by the random walker,…”