MPEJ Volume 3, No.3, 19pp
Received: May 17, 1997, Revised Jun 17, 1997, Accepted: Jun 27, 1997

Pierre Collet, Jean-Pierre Eckmann
Oscillations of Observables in 1-Dimensional Lattice Systems

ABSTRACT: 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 {\em not} depend on any detail of the
model, nor on the state one observes, but only on the ratio $\alpha/\beta$.