Oscillations of Observables in 1-Dimensional Lattice Systems

Using, and extending, striking inequalities by V.V. Ivanov on the down-crossings of monotone functions and ergodic sums, we give universal bounds on the probability of finding oscillations of observables in 1-dimensional lattice gases in infinite volume. In particular, we study the finite volume average of the occupation number as one runs through an increasing sequence of boxes of size $2n$ centered at the origin. We show that the probability to see $k$ oscillations of this average between two values $\beta $ and $0<\alpha <\beta $ is bounded by $C R^k$, with $R<1$, where the constants $C$ and $R$ do not depend on any detail of the model, nor on the state one observes, but only on the ratio $\alpha/\beta $.