A Law of Large Numbers for Random Walks in Random Environment

Multi-excited random walks on integers

“…In the case where ω(x), x ∈ Z d , are i.i.d. under P, a straightforward adaptation of a renewal structure technique introduced by Sznitman and Zerner for random walks in random environments (RWRE) gives the P 0 -a.s. convergence of (X n · ℓ)/n towards a deterministic limit on the event {lim n→∞ X n · ℓ = +∞}, where ℓ is any direction in R d , see [9] and [11, Theorem 3.2.2]. Here we assume that none of the transition probabilities is equal to 0.…”

Random Walks in a Dirichlet Environment

“…We want to apply Sznitman and Zerner's law of large numbers [9]. From a careful reading of the proof of this law of large numbers, we can see that the only conditions that need to be fullfilled, are the integrability of the Green function G ω U (z 0 , z 0 ) for all bounded U , and Kalikow's condition.…”

“…In section 5, we study the integrability of the Green function of the walk which ensures the existence of the original (non killed) Kalikow's auxiliary walk and finish the proof of our first result by applying the law of large numbers of Sznitman and Zerner [9]. In section 6, we follow the scheme of [6] to get bounds for the asymptotic velocity of the walk, and deduce an expansion of the asymptotic velocity at low disorder.…”

“…After a first breakthrough of Kalikow [3], giving a transience criterion for non-reversible multidimensional Random Walks in Random Environment, Sznitman and Zerner proved, several years later, a law of large numbers in [9], followed by a central limit theorem proved by Sznitman in [8].…”

Monotonicity for excited random walk in high dimensions

“…It was shown that this process has ballistic behaviour when d ≥ 2 in [4,12,13]. A nontrivial strong law of large numbers (SLLN) can then be obtained for d ≥ 2 using renewal techniques (see for example [14,16]). For d = 1, it is known that ERW is recurrent and diffusive [7] except in the trivial case β = 1.…”