Abstract
We prove that a law of large numbers and a central limit theorem hold for the excited random walk model in every dimension $d\geq 2$.
Highlights
We prove that a law of large numbers and a central limit theorem hold for the excited random walk model in every dimension d ≥ 2
An excited random walk with bias parameter p ∈ (1/2, 1] is a discrete time nearest neighbor random walk (Xn)n≥0 on the lattice Zd obeying the following rule: when at time n the walk is at a site it has already visited before time n, it jumps uniformly at random to one of the 2d neighboring sites
The excited random walk on a tree was studied by Volkov [15]
Summary
An excited random walk with bias parameter p ∈ (1/2, 1] is a discrete time nearest neighbor random walk (Xn)n≥0 on the lattice Zd obeying the following rule: when at time n the walk is at a site it has already visited before time n, it jumps uniformly at random to one of the 2d neighboring sites. We prove that the biased coordinate of the excited random walk satisfies a law of large numbers and a central limit theorem for every d ≥ 2 and p ∈ Using estimates for the so-called tan points of the simple random walk, introduced in [1] and subsequently used in [7, 8], it is possible to prove that, when d ≥ 2, the number of distinct points visited by the excited random walk after n steps is, with large probability, of order n3/4 at least. Since the excited random walk performs a biased random step at each time it visits a site it has not previously visited, the e1-coordinate of the walk should typically be at least of order n3/4 after n steps Since this number is o(n), this estimate is not good enough to provide a direct proof that the walk has linear speed.
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