Random walks in a moderately sparse random environment

Abstract

A random walk in a sparse random environment is a model introduced by Matzavinos et al. [Electron. J. Probab. 21, paper no. 72: 2016] as a generalization of both a simple symmetric random walk and a classical random walk in a random environment. A random walk (Xn)n∈ℕ∪{0} in a sparse random environment (Sk,λk)k∈ℤ is a nearest neighbor random walk on ℤ that jumps to the left or to the right with probability 1/2 from every point of ℤ\{…, S−1, S0=0, S1,…} and jumps to the right (left) with the random probability λk+1 (1 − λk+1) from the point Sk, k∈ℤ. Assuming that (Sk−Sk−1,λk)k∈ℤ are independent copies of a random vector (ξ, λ)∈ℕ×(0, 1) and the mean Eξ is finite (moderate sparsity) we obtain stable limit laws for Xn, properly normalized and centered, as n → ∞. While the case ξ ≤ M a.s. for some deterministic M > 0 (weak sparsity) was analyzed by Matzavinos et al., the case Eξ=∞ (strong sparsity) will be analyzed in a forthcoming paper.

Critical branching processes in a sparse random environment

“…To close the introduction, we mention that closely related random walks in a sparse random environment, which is an intermediate model between the simple random walk and the random walk in a random environment, have been recently investigated in [4][5][6]. its normalized second factorial moment.…”