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

Abstract

Let {S k , k ≥ 0} be a symmetric random walk on Z d , and {η(x), x ∈ Z d } an independent random field of centered i.i.d. random variables with tail decayWe consider a random walk in random scenery, that is X n = η(S 0 ) + · · · + η(S n ). We present asymptotics for the probability, over both randomness, that {X n > n β } for β > 1/2 and α > 1. To obtain such asymptotics, we establish large deviations estimates for the self-intersection local times process x l 2 n (x), where l n (x) is the number of visits of site x up to time n.

The arcsine law as a universal aging scheme for trap models

“…and heavy tailed? We are not interested here in the case where τ is random but reasonably tame, which is usually studied under the name of Random Walk in Random Scenery (for recent important results see [AC06,GvdHK06,GKS06] and references therein). We will, on the contrary, isolate general conditions bearing both on the distribution of the random landscape τ and on the potential theory of the chain Y which will ensure that the process S(u) can be approximated, in appropriate large time scales, by a stable subordinator.…”

Exponential moments of self-intersection local times of stable random walks in subcritical dimensions

“…we get the following inversion and Parseval's formulas: (2). With this definition, Parseval's identity and inversion formula read…”

“…Rough logarithmic asymptotics in the supercritical case were first proved in [2] (for p = α = 2) and later refined in precise logarithmic asymptotics in [1,26].…”

“…Without loss of generality, we will assume that c = 2π. This choice is made in order to get nice statements using (2) as a definition of Fourier transform.…”

Chover-Type Laws of the Iterated Logarithm for Kesten-Spitzer Random Walks in Random Sceneries Belonging to the Domain of Stable Attraction

“…Finally for any arbitrary transient random walk, it can be shown that the sequence { −1/2 , ∈ N} is asymptotically normal (see for instance Spitzer [7] page 53). Among others, we can cite strong approximation results [8][9][10], laws of the iterated logarithm [11][12][13], limit theorems for correlated sceneries or walks [14][15][16][17], large and moderate deviations results [18][19][20][21][22], and ergodic and mixing properties (see the survey [23]). …”