Moments and Distribution of the Local Times of a Transient Random Walk on ℤ d

Abstract

Consider an arbitrary transient random walk on Z d with d ∈ N. Pick α ∈ [0, ∞) and let L n (α) be the spatial sum of the α-th power of the n-step local times of the walk. Hence, L n (0) is the range, L n (1) = n + 1, and for integers α, L n (α) is the number of the α-fold self-intersections of the walk. We prove a strong law of large numbers for L n (α) as n → ∞. Furthermore, we identify the asymptotic law of the local time in a random site uniformly distributed over the range. These results complement and contrast analogous results for recurrent walks in two dimensions recently derived byČerný [Ce07]. Although these assertions are certainly known to experts, we could find no proof in the literature in this generality.

Optimal Bounds for the Variance of Self-Intersection Local Times

“…The result was motivated by [19] and [3] (and improves related results of Becker and Konig for d = 3 and d = 4). Several cases treated in [2,4,5,8,10,7,3,17] can then be obtained as particular cases.…”

“…In this paper, motivated by the results of Spitzer [19] for genuinely d-dimensional random walks and the approach of Becker and König [3](see also Asselah [2] where non-integer α is also treated) we shall study the asymptotic behavior of var(L n (α)) without imposing any moment assumptions on the random walk. The central idea behind our approach is to compare the selfintersection local times L n (α) of a general d-dimensional walk with those of its symmetrised version.…”

Upper tails of self-intersection local times of random walks: survey of proof techniques

“…(1.4) for some C = C d,p ∈ (0, ∞); see [Ce07] for d = 2 and [BK09] for d ≥ 3. In the following, we will concentrate on d ≥ 2.…”

Shape transition under excess self-intersections for transient random walk

“…Following ideas of Jain and Pruitt [17], and of Le Gall and Rosen [21], Chen obtains a CLT in dimension 3 or more for ||l n || 2 2 . Finally, Becker and König in [6] have shown that for q integer, (i) in d = 3, var(||l n || q q ) ≤ n 3/2 , (ii) in d = 4, var(||l n || q q ) ≤ n log(n), and (iii) in d ≥ 5, var(||l n || q q ) ≤ c d n. Our result deals with the general case (q > 1 real), where no representation of ||l n || q q is possible in terms of multiple time-intersections. We transform Lindeberg's condition into a large deviation event for ||l Tn || 2 2 on the scale of time of the CLT, that is T n ≈ √ n.…”

“…For q positive real, we still call ||l n || q q the q-fold self-intersection local times. In dimension three and more, Becker and König [6] have shown that there are positive constants, say κ(q, d), such that almost surely lim n→∞ ||l n || q q n = κ(q, d).…”