Global observables for random walks: law of large numbers

Abstract

We consider the sums T N = N n=1 F (S n ) where S n is a random walk on Z d and F : Z d → R is a global observable, that is, a bounded function which admits an average value when averaged over large cubes. We show that T N always satisfies the weak Law of Large Numbers but the strong law fails in general except for one dimensional walks with drift. Under additional regularity assumptions on F , we obtain the Strong Law of Large Numbers and estimate the rate of convergence. The growth exponents which we obtain turn out to be optimal in two special cases: for quasiperiodic observables and for random walks in random scenery.

Quantitative global-local mixing for accessible skew products

“…In our setting, the infinite volume average is analogous to statistical infinite volume limits introduced and refined along the years by Van Hove, Fisher [25,Section 3.3] and Ruelle [29, Section 3.9], which built on the inspirational work of Bogoliubov [4]. Global-local mixing has been studied in different situations, for example random walks [23,24], mechanical systems [15,14] and one dimensional parabolic systems [8].…”